基于数据驱动的离散时间深度循环神经网络的耗散系统建模

# Data-driven discrete-time deep neural network-based modeling for dissipative systems
Source: [https://arxiv.org/html/2609.27186](https://arxiv.org/html/2609.27186)
Tuan Luong††thanks:Department of Mechanical Engineering, Sungkyunkwan University, Suwon 16419, South Korea\. E\-mail:luongtuan@g\.skku\.edu\.Hyungpil Moon††thanks:Department of Mechanical Engineering, Sungkyunkwan University, Suwon 16419, South Korea\. E\-mail:hyungpil@g\.skku\.edu\. Corresponding author\.

###### Abstract

Physical AI has gained increasing attention for its role in developing AI systems that better understand, predict, and control real\-world dynamics\. Achieving this requires AI models that not only achieve high prediction accuracy but also preserve fundamental physical properties of dynamical systems\. In this paper, we propose a deep discrete\-time dissipative recurrent neural network \(DissipNet\) that explicitly enforces dissipativity, a key property related to stability and energy dissipation, through structural weight constraints and a dedicated training algorithm\. By construction, the proposed network is capable of learning dissipative dynamics while preserving their inherent stability, which is formally analyzed using Lyapunov theory\. In contrast to Physics\-Informed Neural Networks \(PINNs\), which incorporate governing equations into the training loss but do not guarantee preservation of internal analytical properties such as dissipativity or passivity, our approach provides explicit guarantees on stability at the model level\. We demonstrate the effectiveness of the proposed method through several modeling applications, and compare its performance with a naive recurrent neural network \(RNN\) and a PINN\-based model\.

Keywords:Physical AI, dissipative neural networks, dissipativity, stable system modeling, Lyapunov stability

## 1Introduction

Neural Networks \(NN\) have spread as a powerful tool for multiple purposes in many fields such as signal processing, modelling, and control of dynamical systems\. Many types of NNs, Multi\-Layer Perceptron \(MLP\), Vanilla Recurrent Neural Networks \(RNNs\), Long\-Short Term Memory \(LSTMs\), Convolutional Neural Networks \(CNNs\), Generative Models, etc\., and their hybrid models\[[1](https://arxiv.org/html/2609.27186#bib.bib20)\], have been proposed for different applications\. However, in most of the work, NNs are considered only as a nonlinear system that fit a collection of data, which have no information about the physical characteristics\. In general, a neural network does not preserve the physical properties of a real system\. For example, the modeling of a dynamic system using a traditional data\-based network model might result in a model that violates physical constraints\[[2](https://arxiv.org/html/2609.27186#bib.bib21)\]\. Preserving physical properties of network models, such as stability properties, is therefore an important research topic to be investigated\.

Amongst physical properties, dissipativity is a relevant issue that has been widely used to analyse the stability of linear and nonlinear systems\. The dissipativity theory has found successful applications in various areas such as signal processing\[[3](https://arxiv.org/html/2609.27186#bib.bib3)\], human\-robot interaction\[[4](https://arxiv.org/html/2609.27186#bib.bib18)\], impedance control\[[5](https://arxiv.org/html/2609.27186#bib.bib17)\], disturbance estimation and control of aircraft\[[6](https://arxiv.org/html/2609.27186#bib.bib15)\], mobile robot\[[7](https://arxiv.org/html/2609.27186#bib.bib16)\]\. It may deal with nonlinear systems using only the general characteristics of the input–output dynamics and offers elegant solutions for the proof of absolute stability\. Dissipative systems exhibit many desirable properties, for example, 1\) the free dynamics and zero dynamics of Dissipative systems are Lyapunov stable; 2\) the parallel and negative feedback interconnections of passive systems remain dissipative\. This property allows the independent modelling and control of separate dissipative systems; and 3\) a dissipative system can achieve stability using output feedback\[[8](https://arxiv.org/html/2609.27186#bib.bib14)\]\. The property can be applied to design a controller for the systems that are unstable\. In fact, there are many engineering systems \(e\.g\., high\-performance aircraft or even some more exotic circuits\) which are even designed to be unstable for some range of initial conditions in order to take advantages of improved performance for other ranges of initial data\. For those systems, an input feedforward dissipative \(IFP\) or an output feedback passivity \(OFP\) can be used to render the process dissipative\[[9](https://arxiv.org/html/2609.27186#bib.bib12)\]\.

Due to the importance of dissipative properties, effort has been made to study dissipavivity and passivity \(related property\) of neural networks\. The passivity properties of static multilayer neural networks are examined in\[[10](https://arxiv.org/html/2609.27186#bib.bib4)\]\. By means of analysing the interconnection of error models, the study derived the relationship between passivity and closed\-loop stability\. Passivity properties of dynamic neural networks can be found in\[[11](https://arxiv.org/html/2609.27186#bib.bib5)\]\. In\[[12](https://arxiv.org/html/2609.27186#bib.bib6)\], dynamic multilayer neural networks are used for nonlinear system on\-line identification\. The passivity approach is applied to access several stability properties of the neuro identifier\. The passivity conditions for delayed neural networks \(DNNs\) are considered in\[[13](https://arxiv.org/html/2609.27186#bib.bib11)\], where the passivity conditions for DNNs without uncertainties were derived, and then extend the results to the case with time\-varying parametric uncertainties\. The passivity conditions for DNNs with uncertainties are considered in\[[14](https://arxiv.org/html/2609.27186#bib.bib7)\]\. In\[[8](https://arxiv.org/html/2609.27186#bib.bib14)\], the global dissipativity of a general class of continuous\-time recurrent neural networks are addressed\. It is worth pointing out that although some passivity analysis has been done, they require the solving of linear matrix equations which face challenges when scaling to large networks\. Some recent studies on learning dissipative neural dynamics have been restricted to specific port\-Hamiltonian network structures\[[15](https://arxiv.org/html/2609.27186#bib.bib19)\]\. However, these models are only applicable when the system inputs remain constant\. In\[[16](https://arxiv.org/html/2609.27186#bib.bib1)\], the incremental dissipativity of a system with external inputs is considered\. However, the approach does not consider the state model and requires multi\-step learning process\. In\[[17](https://arxiv.org/html/2609.27186#bib.bib9)\], a continuous dissipative Neural ODE model was obtained by projecting the original ODE onto a space that guarantees dissipativity\. However, while the training process was time\-consuming, a degradation in modeling accuracy may have occurred during the projection step\.

In this paper, we aim to develop a discrete\-time dissipative recurrent neural network \(DissipNet\) for learning a discrete model of a nonlinear system that is known to be dissipative\. Our contribution includes the proposition of a linear matrix inequality condition for the network to ensure its dissipative property\. We also contribute a sufficient learning algorithm that uses simple unconstrained optimization to determine weights that satisfy these conditions, which is simpler and provide more flexibility compared with\[[16](https://arxiv.org/html/2609.27186#bib.bib1)\]\. The network’s stability is proven using Lyapunov theory\. The integration of physical meaning into a generic network aligns with an emerging research topic known as physics\-informed neural network \(PINN\)\[[18](https://arxiv.org/html/2609.27186#bib.bib13)\]\. The underlying approaches are that PINN encode model equations, like Partial Differential Equations \(PDE\), as a component of the neural network itself\. In this approach, NN must fit observed data while reducing a minimize a loss function, which includes reflecting the initial and boundary conditions along the space\-time domain’s boundary and the PDE residual\. It has been demonstrated that the PINN can efficiently leverage the predictive ability of neural networks PINNs addressing problems that are described by few data, or with noisy experiment observations\. The work aims to learn the equations and replace them by NNs\. However, there is no guarantee that the NN will preserve the equation’s internal analytical properties, for example, passivity in case of robotics, as opposed to our proposal\.

The remaining of this paper is organized as follows\. Section[2](https://arxiv.org/html/2609.27186#S2)presents the dissipative neural network model and its learning algorithm\. Case studies demonstrating the performance of the proposed network and comparisons with a PINN model and a Naive RNN model are presented in Section[3](https://arxiv.org/html/2609.27186#S3)\. We conclude the paper in Section[5](https://arxiv.org/html/2609.27186#S5)\.

## 2Dissipative Recurrent Neural Networks

In this section, the problem under research will be described, and the proposed network and the conditions for it to be dissipative will be given\. Then, an algorithm on how the neural network will be trained will be presented\.

### 2\.1Problem formulation

In this work, we consider the discrete nonlinear systems of the form

𝒙k\+1=𝒇⁡\(𝒙k,𝒖k\)\\mbox\{\\boldmath$x$\}\_\{k\+1\}=\\mbox\{\\boldmath$f$\}\(\\mbox\{\\boldmath$x$\}\_\{k\},\\mbox\{\\boldmath$u$\}\_\{k\}\)\(1\)𝒚k=𝒈⁡\(𝒙k,𝒖k\)\\mbox\{\\boldmath$y$\}\_\{k\}=\\mbox\{\\boldmath$g$\}\(\\mbox\{\\boldmath$x$\}\_\{k\},\\mbox\{\\boldmath$u$\}\_\{k\}\)\(2\)where𝒙k∈ℝn×n,𝒖k∈ℝm×1,𝒚k∈ℝp×1\\mbox\{\\boldmath$x$\}\_\{k\}\\in\\mathbb\{R\}^\{n\\times n\},\\mbox\{\\boldmath$u$\}\_\{k\}\\in\\mathbb\{R\}^\{m\\times 1\},\\mbox\{\\boldmath$y$\}\_\{k\}\\in\\mathbb\{R\}^\{p\\times 1\}are states, inputs and outputs of the system at time stepkk\.𝒇⁡\(0\)=0\\mbox\{\\boldmath$f$\}\(0\)=0,𝒈⁡\(0\)=0\\mbox\{\\boldmath$g$\}\(0\)=0\.

###### Definition 1\.

Definition 1: The system in Eq\.\([1](https://arxiv.org/html/2609.27186#S2.E1)\) and Eq\.\([2](https://arxiv.org/html/2609.27186#S2.E2)\) is said to be\(𝐐,𝐒,𝐑\)\(\\mbox\{\\boldmath$Q$\},\\mbox\{\\boldmath$S$\},\\mbox\{\\boldmath$R$\}\)dissipative, where0⪰𝐐∈ℝp×p0\\succeq\\mbox\{\\boldmath$Q$\}\\in\\mathbb\{R\}^\{p\\times p\},𝐒∈ℝm×p\\mbox\{\\boldmath$S$\}\\in\\mathbb\{R\}^\{m\\times p\}and𝐑=𝐑T∈ℝm×m\\mbox\{\\boldmath$R$\}=\\mbox\{\\boldmath$R$\}^\{T\}\\in\\mathbb\{R\}^\{m\\times m\}if there exists a Lyapunov functionVk​\(𝐱k\)≥0V\_\{k\}\(\\mbox\{\\boldmath$x$\}\_\{k\}\)\\geq 0such that

Vk\+1​\(𝒙k\+1\)−Vk​\(𝒙k\)≤\[𝒚kT𝒖kT\]​\[𝑸𝑺T𝑺𝑹\]​\[𝒚k𝒖k\]V\_\{k\+1\}\(\\mbox\{\\boldmath$x$\}\_\{k\+1\}\)\-V\_\{k\}\(\\mbox\{\\boldmath$x$\}\_\{k\}\)\\leq\[\\mbox\{\\boldmath$y$\}\_\{k\}^\{T\}\\quad\\mbox\{\\boldmath$u$\}\_\{k\}^\{T\}\]\\begin\{bmatrix\}\\mbox\{\\boldmath$Q$\}&\\mbox\{\\boldmath$S$\}^\{T\}\\\\ \\mbox\{\\boldmath$S$\}&\\mbox\{\\boldmath$R$\}\\end\{bmatrix\}\\begin\{bmatrix\}\\mbox\{\\boldmath$y$\}\_\{k\}\\\\ \\mbox\{\\boldmath$u$\}\_\{k\}\\end\{bmatrix\}\(3\)

Some important cases of the\(𝑸,𝑺,𝑹\)\(\\mbox\{\\boldmath$Q$\},\\mbox\{\\boldmath$S$\},\\mbox\{\\boldmath$R$\}\)dissipative are found in passivity, finite\-gainℒ2\\mathcal\{L\}\_\{2\}stability\. A passive system is dissipative with supply rate given by𝑸=𝟎,𝑹=𝟎\\mbox\{\\boldmath$Q$\}=\\mbox\{\\boldmath$0$\},\\mbox\{\\boldmath$R$\}=\\mbox\{\\boldmath$0$\}, and𝑺=12​𝑰\\mbox\{\\boldmath$S$\}=\\frac\{1\}\{2\}\\mbox\{\\boldmath$I$\}\. Anℒ2\\mathcal\{L\}\_\{2\}system is dissipative with supply rate𝑺=𝟎,𝑸=1ξ​𝑰\\mbox\{\\boldmath$S$\}=\\mbox\{\\boldmath$0$\},\\mbox\{\\boldmath$Q$\}=\\frac\{1\}\{\\xi\}\\mbox\{\\boldmath$I$\}, whereξ\\xiis theℒ2\\mathcal\{L\}\_\{2\}gain of the system\. When the input\-output data\(𝒖k,𝒚k\)\(\\mbox\{\\boldmath$u$\}\_\{k\},\\mbox\{\\boldmath$y$\}\_\{k\}\)are given and are modeled using RNN networks, the system dynamics is represented by the following equations:

𝒉k\+1=fr​n​n​\(𝒉k,𝒖k,θr​n​n\)\\mbox\{\\boldmath$h$\}\_\{k\+1\}=f\_\{rnn\}\(\\mbox\{\\boldmath$h$\}\_\{k\},\\mbox\{\\boldmath$u$\}\_\{k\},\\theta\_\{rnn\}\)\(4\)𝒚^k=𝑪𝒉kL\+𝑫𝒖k\+𝒃y\\hat\{\\mbox\{\\boldmath$y$\}\}\_\{k\}=\\mbox\{\\boldmath$C$\}\\mbox\{\\boldmath$h$\}^\{L\}\_\{k\}\+\\mbox\{\\boldmath$D$\}\\mbox\{\\boldmath$u$\}\_\{k\}\+\\mbox\{\\boldmath$b$\}\_\{y\}\(5\)minθr​n​n,𝑪,𝑫∑i=1Nℒ⁡\(𝒚^i,𝒚i\)\\min\_\{\\begin\{subarray\}\{c\}\\theta\_\{rnn\},\\mbox\{\\boldmath$C$\},\\mbox\{\\boldmath$D$\}\\end\{subarray\}\}\\quad\\sum\_\{i=1\}^\{N\}\\mathcal\{L\}\(\\hat\{\\mbox\{\\boldmath$y$\}\}\_\{i\},\\mbox\{\\boldmath$y$\}\_\{i\}\)\(6\)whereθr​n​n\\theta\_\{rnn\}are trainable weights of the RNN network with L layers,𝒉k=\[𝒉k0T𝒉k1T⋯𝒉kLT\]T\\mbox\{\\boldmath$h$\}\_\{k\}=\[\{\\mbox\{\\boldmath$h$\}^\{0\}\_\{k\}\}^\{T\}\\quad\{\\mbox\{\\boldmath$h$\}^\{1\}\_\{k\}\}^\{T\}\\quad\\cdots\\quad\{\\mbox\{\\boldmath$h$\}^\{L\}\_\{k\}\}^\{T\}\]^\{T\},𝒉ki∈ℝq×1\(i=1,⋯,L\)\\mbox\{\\boldmath$h$\}^\{i\}\_\{k\}\\in\\mathbb\{R\}^\{q\\times 1\}\\quad\(i=1,\\cdots,L\)are the vectors of hidden states,𝑪∈ℝp×q,𝑫∈ℝp×m\\mbox\{\\boldmath$C$\}\\in\\mathbb\{R\}^\{p\\times q\},\\mbox\{\\boldmath$D$\}\\in\\mathbb\{R\}^\{p\\times m\}, and𝒚^i,𝒚i\\hat\{\\mbox\{\\boldmath$y$\}\}\_\{i\},\\mbox\{\\boldmath$y$\}\_\{i\}are the output of the model and the ground truth, respectively at the time stepi⁡\(i=1,⋯,N\)i\(i=1,\\cdots,N\), where N is the number of data samples\. If we choose𝒙k=𝒉k\\mbox\{\\boldmath$x$\}\_\{k\}=\\mbox\{\\boldmath$h$\}\_\{k\}, the Eqs\. \([4](https://arxiv.org/html/2609.27186#S2.E4)\-[5](https://arxiv.org/html/2609.27186#S2.E5)\) have the same form as Eqs\. \([1](https://arxiv.org/html/2609.27186#S2.E1)\-[2](https://arxiv.org/html/2609.27186#S2.E2)\)\. Note that for a discrete\-time system to be dissipative,D≠𝟎D\\neq\\mbox\{\\boldmath$0$\}is necessary\[[19](https://arxiv.org/html/2609.27186#bib.bib2)\]\.

### 2\.2Dissipative Recurrent Neural Networks

The model in Eq\. \([4](https://arxiv.org/html/2609.27186#S2.E4)\) can be expanded in the following form

𝒉k\+11\\displaystyle\\mbox\{\\boldmath$h$\}^\{1\}\_\{k\+1\}=ϕ⁡\(𝑾i​h1​𝒖k\+𝑾h​h1​𝒉k1\+𝒃h1\)\\displaystyle=\\phi\(\\mbox\{\\boldmath$W$\}^\{1\}\_\{ih\}\\mbox\{\\boldmath$u$\}\_\{k\}\+\\mbox\{\\boldmath$W$\}^\{1\}\_\{hh\}\\mbox\{\\boldmath$h$\}^\{1\}\_\{k\}\+\\mbox\{\\boldmath$b$\}^\{1\}\_\{h\}\)\(7\)𝒉k\+12\\displaystyle\\mbox\{\\boldmath$h$\}^\{2\}\_\{k\+1\}=ϕ⁡\(𝑾i​h2​𝒉k\+11\+𝑾h​h2​𝒉k2\+𝒃h2\)\\displaystyle=\\phi\(\\mbox\{\\boldmath$W$\}^\{2\}\_\{ih\}\\mbox\{\\boldmath$h$\}^\{1\}\_\{k\+1\}\+\\mbox\{\\boldmath$W$\}^\{2\}\_\{hh\}\\mbox\{\\boldmath$h$\}^\{2\}\_\{k\}\+\\mbox\{\\boldmath$b$\}^\{2\}\_\{h\}\)𝒉k\+1L\\displaystyle\\mbox\{\\boldmath$h$\}^\{L\}\_\{k\+1\}=ϕ⁡\(𝑾i​hL​𝒉k\+1L−1\+𝑾h​hL​𝒉kL\+𝒃hL\)\\displaystyle=\\phi\(\\mbox\{\\boldmath$W$\}^\{L\}\_\{ih\}\\mbox\{\\boldmath$h$\}^\{L\-1\}\_\{k\+1\}\+\\mbox\{\\boldmath$W$\}^\{L\}\_\{hh\}\\mbox\{\\boldmath$h$\}^\{L\}\_\{k\}\+\\mbox\{\\boldmath$b$\}^\{L\}\_\{h\}\)𝒚^k=𝑪𝒉kL\+𝑫𝒖k\+𝒃y\\hat\{\\mbox\{\\boldmath$y$\}\}\_\{k\}=\\mbox\{\\boldmath$C$\}\\mbox\{\\boldmath$h$\}^\{L\}\_\{k\}\+\\mbox\{\\boldmath$D$\}\\mbox\{\\boldmath$u$\}\_\{k\}\+\\mbox\{\\boldmath$b$\}\_\{y\}\(8\)where𝑾i​h1∈ℝq×m\\mbox\{\\boldmath$W$\}^\{1\}\_\{ih\}\\in\\mathbb\{R\}^\{q\\times m\},𝒃hiT∈ℝq×q,𝑾h​hi∈ℝq×q\(i=1,⋯,L\)\{\\mbox\{\\boldmath$b$\}^\{i\}\_\{h\}\}^\{T\}\\in\\mathbb\{R\}^\{q\\times q\},\\mbox\{\\boldmath$W$\}^\{i\}\_\{hh\}\\in\\mathbb\{R\}^\{q\\times q\}\\quad\(i=1,\\cdots,L\),𝑾i​hi∈ℝq×q\(i=2,⋯,L\)\\mbox\{\\boldmath$W$\}^\{i\}\_\{ih\}\\in\\mathbb\{R\}^\{q\\times q\}\\quad\(i=2,\\cdots,L\)and𝑩h∈ℝn×q\\mbox\{\\boldmath$B$\}\_\{h\}\\in\\mathbb\{R\}^\{n\\times q\},𝑾i​h1∈ℝq×m\\mbox\{\\boldmath$W$\}^\{1\}\_\{ih\}\\in\\mathbb\{R\}^\{q\\times m\}\.ϕ⁡\(⋅\)\\phi\(\\cdot\)is the activation function\.

Eq\. \([7](https://arxiv.org/html/2609.27186#S2.E7)\) can be represented as follows

𝒉k\+1=ϕ⁡\(𝑾u​𝒖k\+𝑾i​h​𝒉k\+1\+𝑾h​h​𝒉k\+𝒃h\)\\mbox\{\\boldmath$h$\}\_\{k\+1\}=\\phi\(\\mbox\{\\boldmath$W$\}\_\{u\}\\mbox\{\\boldmath$u$\}\_\{k\}\+\\mbox\{\\boldmath$W$\}\_\{ih\}\\mbox\{\\boldmath$h$\}\_\{k\+1\}\+\\mbox\{\\boldmath$W$\}\_\{hh\}\\mbox\{\\boldmath$h$\}\_\{k\}\+\\mbox\{\\boldmath$b$\}\_\{h\}\)\(9\)𝒚^k=𝑪h​𝒉k\+𝑫𝒖k\+𝒃y\\hat\{\\mbox\{\\boldmath$y$\}\}\_\{k\}=\\mbox\{\\boldmath$C$\}\_\{h\}\\mbox\{\\boldmath$h$\}\_\{k\}\+\\mbox\{\\boldmath$D$\}\\mbox\{\\boldmath$u$\}\_\{k\}\+\\mbox\{\\boldmath$b$\}\_\{y\}\(10\)Where

𝑾h​h=\[𝑾h​h1𝟎⋯𝟎𝟎𝑾h​h2⋯𝟎⋱𝟎𝟎⋯𝑾h​hL\]∈ℝL​q×L​q\\mbox\{\\boldmath$W$\}\_\{hh\}=\\begin\{bmatrix\}\\mbox\{\\boldmath$W$\}^\{1\}\_\{hh\}&\\mbox\{\\boldmath$0$\}&\\cdots&\\mbox\{\\boldmath$0$\}\\\\ \\mbox\{\\boldmath$0$\}&\\mbox\{\\boldmath$W$\}^\{2\}\_\{hh\}&\\cdots&\\mbox\{\\boldmath$0$\}\\\\ \\vdots&\\vdots&\\ddots&\\vdots\\\\ \\mbox\{\\boldmath$0$\}&\\mbox\{\\boldmath$0$\}&\\cdots&\\mbox\{\\boldmath$W$\}^\{L\}\_\{hh\}\\\\ \\end\{bmatrix\}\\in\\mathbb\{R\}^\{Lq\\times Lq\}\(11\)𝑾i​h=\[𝟎𝟎⋯𝟎𝑾i​h2𝟎⋯𝟎⋱𝟎𝟎𝑾i​hL𝟎\]∈ℝL​q×L​q\\mbox\{\\boldmath$W$\}\_\{ih\}=\\begin\{bmatrix\}\\mbox\{\\boldmath$0$\}&\\mbox\{\\boldmath$0$\}&\\cdots&\\mbox\{\\boldmath$0$\}\\\\ \\mbox\{\\boldmath$W$\}^\{2\}\_\{ih\}&\\mbox\{\\boldmath$0$\}&\\cdots&\\mbox\{\\boldmath$0$\}\\\\ \\vdots&\\vdots&\\ddots&\\vdots\\\\ \\mbox\{\\boldmath$0$\}&\\mbox\{\\boldmath$0$\}&\\mbox\{\\boldmath$W$\}^\{L\}\_\{ih\}&\\mbox\{\\boldmath$0$\}\\\\ \\end\{bmatrix\}\\in\\mathbb\{R\}^\{Lq\\times Lq\}\(12\)𝑾u=\[𝑾i​h1𝟎𝟎𝟎\]∈ℝL​q×m\\mbox\{\\boldmath$W$\}\_\{u\}=\\begin\{bmatrix\}\\mbox\{\\boldmath$W$\}^\{1\}\_\{ih\}\\\\ \\mbox\{\\boldmath$0$\}\\\\ \\mbox\{\\boldmath$0$\}\\\\ \\vdots\\\\ \\mbox\{\\boldmath$0$\}\\\\ \\end\{bmatrix\}\\in\\mathbb\{R\}^\{Lq\\times m\}\(13\)𝑪h=\[𝑪𝟎𝟎⋯𝟎\]∈ℝp×L​q\\mbox\{\\boldmath$C$\}\_\{h\}=\\begin\{bmatrix\}\\mbox\{\\boldmath$C$\}&\\mbox\{\\boldmath$0$\}&\\mbox\{\\boldmath$0$\}\\cdots&\\mbox\{\\boldmath$0$\}\\end\{bmatrix\}\\in\\mathbb\{R\}^\{p\\times Lq\}\(14\)𝒃h=\[𝒃h1T𝒃h2T⋯𝒃hLT\]T∈ℝL​q×1\\mbox\{\\boldmath$b$\}\_\{h\}=\[\{\\mbox\{\\boldmath$b$\}^\{1\}\_\{h\}\}^\{T\}\\quad\{\\mbox\{\\boldmath$b$\}^\{2\}\_\{h\}\}^\{T\}\\quad\\cdots\\quad\{\\mbox\{\\boldmath$b$\}^\{L\}\_\{h\}\}^\{T\}\]^\{T\}\\in\\mathbb\{R\}^\{Lq\\times 1\}\(15\)𝒃y∈ℝp×1\\mbox\{\\boldmath$b$\}\_\{y\}\\in\\mathbb\{R\}^\{p\\times 1\}\(16\)For a sequence length of T, the RNN network can be represented as

𝒉k\+1\\displaystyle\\mbox\{\\boldmath$h$\}\_\{k\+1\}=ϕ⁡\(𝑾u​𝒖k\+𝑾i​h​𝒉k\+1\+𝑾h​h​𝒉k\+𝒃h\)\\displaystyle=\\phi\(\\mbox\{\\boldmath$W$\}\_\{u\}\\mbox\{\\boldmath$u$\}\_\{k\}\+\\mbox\{\\boldmath$W$\}\_\{ih\}\\mbox\{\\boldmath$h$\}\_\{k\+1\}\+\\mbox\{\\boldmath$W$\}\_\{hh\}\\mbox\{\\boldmath$h$\}\_\{k\}\+\\mbox\{\\boldmath$b$\}\_\{h\}\)\(17\)𝒉k\+2\\displaystyle\\mbox\{\\boldmath$h$\}\_\{k\+2\}=ϕ⁡\(𝑾u​𝒖k\+1\+𝑾i​h​𝒉k\+2\+𝑾h​h​𝒉k\+1\+𝒃h\)\\displaystyle=\\phi\(\\mbox\{\\boldmath$W$\}\_\{u\}\\mbox\{\\boldmath$u$\}\_\{k\+1\}\+\\mbox\{\\boldmath$W$\}\_\{ih\}\\mbox\{\\boldmath$h$\}\_\{k\+2\}\+\\mbox\{\\boldmath$W$\}\_\{hh\}\\mbox\{\\boldmath$h$\}\_\{k\+1\}\+\\mbox\{\\boldmath$b$\}\_\{h\}\)𝒉k\+T\\displaystyle\\mbox\{\\boldmath$h$\}\_\{k\+T\}=ϕ⁡\(𝑾u​𝒖k\+T\+𝑾i​h​𝒉k\+T\+𝑾h​h​𝒉k\+T−1\+𝒃h\)\\displaystyle=\\phi\(\\mbox\{\\boldmath$W$\}\_\{u\}\\mbox\{\\boldmath$u$\}\_\{k\+T\}\+\\mbox\{\\boldmath$W$\}\_\{ih\}\\mbox\{\\boldmath$h$\}\_\{k\+T\}\+\\mbox\{\\boldmath$W$\}\_\{hh\}\\mbox\{\\boldmath$h$\}\_\{k\+T\-1\}\+\\mbox\{\\boldmath$b$\}\_\{h\}\)Then

𝒚^k\+T−1=𝑪h​𝒉k\+T−1L\+𝑫𝒖k\+T−1\+𝒃y\\hat\{\\mbox\{\\boldmath$y$\}\}\_\{k\+T\-1\}=\\mbox\{\\boldmath$C$\}\_\{h\}\\mbox\{\\boldmath$h$\}^\{L\}\_\{k\+T\-1\}\+\\mbox\{\\boldmath$D$\}\\mbox\{\\boldmath$u$\}\_\{k\+T\-1\}\+\\mbox\{\\boldmath$b$\}\_\{y\}\(18\)Eqs\. \([17](https://arxiv.org/html/2609.27186#S2.E17)\)\-\([18](https://arxiv.org/html/2609.27186#S2.E18)\) can be converted into the form of Eqs\. \([1](https://arxiv.org/html/2609.27186#S2.E1)\)\-\([2](https://arxiv.org/html/2609.27186#S2.E2)\) as follows

𝝂k\+1\\displaystyle\\bm\{\\nu\}\_\{k\+1\}=ϕ1​\(𝑾u​𝒖k\+𝑾i​h​𝝂k\+1\+𝑾h​h​𝝂k\)\\displaystyle=\\phi\_\{1\}\(\\mbox\{\\boldmath$W$\}\_\{u\}\\mbox\{\\boldmath$u$\}\_\{k\}\+\\mbox\{\\boldmath$W$\}\_\{ih\}\\bm\{\\nu\}\_\{k\+1\}\+\\mbox\{\\boldmath$W$\}\_\{hh\}\\bm\{\\nu\}\_\{k\}\)\(19\)𝝂k\+2\\displaystyle\\bm\{\\nu\}\_\{k\+2\}=ϕ1​\(𝑾u​𝒖k\+1\+𝑾i​h​𝝂k\+2\+𝑾h​h​𝝂k\+1\)\\displaystyle=\\phi\_\{1\}\(\\mbox\{\\boldmath$W$\}\_\{u\}\\mbox\{\\boldmath$u$\}\_\{k\+1\}\+\\mbox\{\\boldmath$W$\}\_\{ih\}\\bm\{\\nu\}\_\{k\+2\}\+\\mbox\{\\boldmath$W$\}\_\{hh\}\\bm\{\\nu\}\_\{k\+1\}\)𝝂k\+T\\displaystyle\\bm\{\\nu\}\_\{k\+T\}=ϕ1​\(𝑾u​𝒖k\+T\+𝑾i​h​𝝂k\+T\+𝑾h​h​𝝂k\+T−1\)\\displaystyle=\\phi\_\{1\}\(\\mbox\{\\boldmath$W$\}\_\{u\}\\mbox\{\\boldmath$u$\}\_\{k\+T\}\+\\mbox\{\\boldmath$W$\}\_\{ih\}\\bm\{\\nu\}\_\{k\+T\}\+\\mbox\{\\boldmath$W$\}\_\{hh\}\\bm\{\\nu\}\_\{k\+T\-1\}\)Then

𝒚^k\+T−1=𝑪h​𝝂k\+T−1L\+𝑫𝒖k\+T−1\\hat\{\\mbox\{\\boldmath$y$\}\}\_\{k\+T\-1\}=\\mbox\{\\boldmath$C$\}\_\{h\}\\bm\{\\nu\}^\{L\}\_\{k\+T\-1\}\+\\mbox\{\\boldmath$D$\}\\mbox\{\\boldmath$u$\}\_\{k\+T\-1\}\(20\)where𝝂k=𝒉k−ϕ⁡\(𝟎\),ϕ1=ϕ−ϕ⁡\(𝟎\),𝒃h=\(𝑾i​h\+𝑾h​h\)​ϕ​\(𝟎\),\\bm\{\\nu\}\_\{k\}=\\mbox\{\\boldmath$h$\}\_\{k\}\-\\phi\(\\mbox\{\\boldmath$0$\}\),\\phi\_\{1\}=\\phi\-\\phi\(\\mbox\{\\boldmath$0$\}\),\\mbox\{\\boldmath$b$\}\_\{h\}=\(\\mbox\{\\boldmath$W$\}\_\{ih\}\+\\mbox\{\\boldmath$W$\}\_\{hh\}\)\\phi\(\\mbox\{\\boldmath$0$\}\),𝒃y=𝑪h​ϕ​\(𝟎\),\\mbox\{\\boldmath$b$\}\_\{y\}=\\mbox\{\\boldmath$C$\}\_\{h\}\\phi\(\\mbox\{\\boldmath$0$\}\),

###### Assumption 1\.

The activation functionθ\\thetais piecewise differentiable and has slope restricted in\[α=0,β\]\[\\alpha=0,\\beta\]

α=0≤ϕ⁡\(x2\)−ϕ⁡\(x1\)x2−x1≤β\\alpha=0\\leq\\frac\{\\phi\(x\_\{2\}\)\-\\phi\(x\_\{1\}\)\}\{x\_\{2\}\-x\_\{1\}\}\\leq\\beta\(21\)

This assumption holds for the most widely used activation functions, such as tanh, ReLU, and the sigmoid function\. While the dissipativity condition proposed in this work can be easily extended for all finite values ofα\\alphaandβ\\beta, the proposed training algorithm is restricted to the caseα=0\\alpha=0\. The extension to cases withα≠0\\alpha\\neq 0is left for future work\.

###### Proposition 1\.

The systems described in Eqs\. \([19](https://arxiv.org/html/2609.27186#S2.E19)\)\-\([20](https://arxiv.org/html/2609.27186#S2.E20)\) will be\(𝐐,𝐒,𝐑\)\(\\mbox\{\\boldmath$Q$\},\\mbox\{\\boldmath$S$\},\\mbox\{\\boldmath$R$\}\)dissipative if there is a positive definite matrix𝐏=𝐏T\\mbox\{\\boldmath$P$\}=\\mbox\{\\boldmath$P$\}^\{T\}such that

\[𝑷\+𝑪hT​𝐐𝐂h−γ​𝑾h​hT​𝚲𝑪hT​𝑺T\+𝑪hT​𝐐𝐃−γ​𝚲​𝑾h​h𝚲−γ​𝑾i​h∗−𝑷−γ​𝚲​𝑾u𝐒𝐂h\+𝑫T​𝑸​Ch−γ​𝑾uT​𝚲𝑹1\]\\begin\{bmatrix\}\\mbox\{\\boldmath$P$\}\+\\mbox\{\\boldmath$C$\}\_\{h\}^\{T\}\\mbox\{\\boldmath$Q$\}\\mbox\{\\boldmath$C$\}\_\{h\}&\-\\gamma\\mbox\{\\boldmath$W$\}\_\{hh\}^\{T\}\\bm\{\\Lambda\}&\\mbox\{\\boldmath$C$\}\_\{h\}^\{T\}\\mbox\{\\boldmath$S$\}^\{T\}\+\\mbox\{\\boldmath$C$\}\_\{h\}^\{T\}\\mbox\{\\boldmath$Q$\}\\mbox\{\\boldmath$D$\}\\\\ \-\\gamma\\bm\{\\Lambda\}\\mbox\{\\boldmath$W$\}\_\{hh\}&\\bm\{\\Lambda\}\-\\gamma\\mbox\{\\boldmath$W$\}^\{\*\}\_\{ih\}\-\\mbox\{\\boldmath$P$\}&\-\\gamma\\bm\{\\Lambda\}\\mbox\{\\boldmath$W$\}\_\{u\}\\\\ \\mbox\{\\boldmath$S$\}\\mbox\{\\boldmath$C$\}\_\{h\}\+\\mbox\{\\boldmath$D$\}^\{T\}\\mbox\{\\boldmath$Q$\}C\_\{h\}&\-\\gamma\\mbox\{\\boldmath$W$\}\_\{u\}^\{T\}\\bm\{\\Lambda\}&\\mbox\{\\boldmath$R$\}\_\{1\}\\end\{bmatrix\}\\\\\(22\)is positive definite\.

where𝐖i​h∗=𝚲​𝐖i​h\+𝐖i​hT​𝚲\\mbox\{\\boldmath$W$\}^\{\*\}\_\{ih\}=\\bm\{\\Lambda\}\\mbox\{\\boldmath$W$\}\_\{ih\}\+\\mbox\{\\boldmath$W$\}\_\{ih\}^\{T\}\\bm\{\\Lambda\},𝐑1=𝐑\+𝐒𝐃\+𝐃T​𝐒T\+𝐃T​𝐐𝐃\\mbox\{\\boldmath$R$\}\_\{1\}=\\mbox\{\\boldmath$R$\}\+\\mbox\{\\boldmath$S$\}\\mbox\{\\boldmath$D$\}\+\\mbox\{\\boldmath$D$\}^\{T\}\\mbox\{\\boldmath$S$\}^\{T\}\+\\mbox\{\\boldmath$D$\}^\{T\}\\mbox\{\\boldmath$Q$\}\\mbox\{\\boldmath$D$\},𝚲=d​i​a​g​\(λ1,λ2,⋯,λL​q\)\\bm\{\\Lambda\}=diag\(\\lambda\_\{1\},\\lambda\_\{2\},\\cdots,\\lambda\_\{Lq\}\)withλi\>0\(i=1,⋯,L​q\)\\lambda\_\{i\}\>0\\quad\(i=1,\\cdots,Lq\)\.

###### Proof\.

Consider the Lyapunov function as follows

Vk​\(𝝂k\)=𝝂kT​𝑷​𝝂kV\_\{k\}\(\\bm\{\\nu\}\_\{k\}\)=\\bm\{\\nu\}\_\{k\}^\{T\}\\mbox\{\\boldmath$P$\}\\bm\{\\nu\}\_\{k\}\(23\)Left and right multiplying Eq\. \([22](https://arxiv.org/html/2609.27186#S2.E22)\) by\[𝝂kT𝝂k\+1T𝒖kT\]\[\\bm\{\\nu\}\_\{k\}^\{T\}\\quad\\bm\{\\nu\}\_\{k\+1\}^\{T\}\\quad\\mbox\{\\boldmath$u$\}\_\{k\}^\{T\}\]and\[𝝂kT𝝂k\+1T𝒖kT\]T\[\\bm\{\\nu\}\_\{k\}^\{T\}\\quad\\bm\{\\nu\}\_\{k\+1\}^\{T\}\\quad\\mbox\{\\boldmath$u$\}\_\{k\}^\{T\}\]^\{T\}, where𝒗k=𝑾u​𝒖k\+𝑾i​h​𝝂k\+1\+𝑾h​h​𝝂k\\mbox\{\\boldmath$v$\}\_\{k\}=\\mbox\{\\boldmath$W$\}\_\{u\}\\mbox\{\\boldmath$u$\}\_\{k\}\+\\mbox\{\\boldmath$W$\}\_\{ih\}\\bm\{\\nu\}\_\{k\+1\}\+\\mbox\{\\boldmath$W$\}\_\{hh\}\\bm\{\\nu\}\_\{k\}, respectively, one can obtain \[see Appendices[A](https://arxiv.org/html/2609.27186#A1)\]\.

Vk\+1​\(𝝂k\+1\)−Vk​\(𝝂k\)≤\[𝒚kT𝒖kT\]​\[𝑸𝑺T𝑺𝑹\]​\[𝒚k𝒖k\]\+\[𝒗kT𝝂k\+1T\]​\[𝟎−γ​𝚲−γ​𝚲𝚲\]​\[𝒗k𝝂k\+1\]≤\[𝒚kT𝒖kT\]​\[𝑸𝑺T𝑺𝑹\]​\[𝒚k𝒖k\]V\_\{k\+1\}\(\\bm\{\\nu\}\_\{k\+1\}\)\-V\_\{k\}\(\\bm\{\\nu\}\_\{k\}\)\\leq\[\\mbox\{\\boldmath$y$\}\_\{k\}^\{T\}\\quad\\mbox\{\\boldmath$u$\}\_\{k\}^\{T\}\]\\begin\{bmatrix\}\\mbox\{\\boldmath$Q$\}&\\mbox\{\\boldmath$S$\}^\{T\}\\\\ \\mbox\{\\boldmath$S$\}&\\mbox\{\\boldmath$R$\}\\end\{bmatrix\}\\begin\{bmatrix\}\\mbox\{\\boldmath$y$\}\_\{k\}\\\\ \\mbox\{\\boldmath$u$\}\_\{k\}\\end\{bmatrix\}\\\\ \+\[\\mbox\{\\boldmath$v$\}\_\{k\}^\{T\}\\quad\\bm\{\\nu\}\_\{k\+1\}^\{T\}\]\\begin\{bmatrix\}\\mbox\{\\boldmath$0$\}&\-\\gamma\\bm\{\\Lambda\}\\\\ \-\\gamma\\bm\{\\Lambda\}&\\bm\{\\Lambda\}\\end\{bmatrix\}\\begin\{bmatrix\}\\mbox\{\\boldmath$v$\}\_\{k\}\\\\ \\bm\{\\nu\}\_\{k\+1\}\\end\{bmatrix\}\\\\ \\leq\[\\mbox\{\\boldmath$y$\}\_\{k\}^\{T\}\\quad\\mbox\{\\boldmath$u$\}\_\{k\}^\{T\}\]\\begin\{bmatrix\}\\mbox\{\\boldmath$Q$\}&\\mbox\{\\boldmath$S$\}^\{T\}\\\\ \\mbox\{\\boldmath$S$\}&\\mbox\{\\boldmath$R$\}\\end\{bmatrix\}\\begin\{bmatrix\}\\mbox\{\\boldmath$y$\}\_\{k\}\\\\ \\mbox\{\\boldmath$u$\}\_\{k\}\\end\{bmatrix\}\(24\)Moreover, it can be shown that \[see Appendices[B](https://arxiv.org/html/2609.27186#A2)\]

\[𝒗kT𝝂k\+1T\]​\[𝟎−γ​𝚲−γ​𝚲𝚲\]​\[𝒗k𝝂k\+1\]≤0\[\\mbox\{\\boldmath$v$\}\_\{k\}^\{T\}\\quad\\bm\{\\nu\}\_\{k\+1\}^\{T\}\]\\begin\{bmatrix\}\\mbox\{\\boldmath$0$\}&\-\\gamma\\bm\{\\Lambda\}\\\\ \-\\gamma\\bm\{\\Lambda\}&\\bm\{\\Lambda\}\\end\{bmatrix\}\\begin\{bmatrix\}\\mbox\{\\boldmath$v$\}\_\{k\}\\\\ \\bm\{\\nu\}\_\{k\+1\}\\end\{bmatrix\}\\leq 0\(25\)From Eq\. \([24](https://arxiv.org/html/2609.27186#S2.E24)\) and Eq\. \([25](https://arxiv.org/html/2609.27186#S2.E25)\), one has

Vk\+1​\(𝝂k\+1\)−Vk​\(𝝂k\)≤\[𝒚kT𝒖kT\]​\[𝑸𝑺T𝑺𝑹\]​\[𝒚k𝒖k\]V\_\{k\+1\}\(\\bm\{\\nu\}\_\{k\+1\}\)\-V\_\{k\}\(\\bm\{\\nu\}\_\{k\}\)\\leq\[\\mbox\{\\boldmath$y$\}\_\{k\}^\{T\}\\quad\\mbox\{\\boldmath$u$\}\_\{k\}^\{T\}\]\\begin\{bmatrix\}\\mbox\{\\boldmath$Q$\}&\\mbox\{\\boldmath$S$\}^\{T\}\\\\ \\mbox\{\\boldmath$S$\}&\\mbox\{\\boldmath$R$\}\\end\{bmatrix\}\\begin\{bmatrix\}\\mbox\{\\boldmath$y$\}\_\{k\}\\\\ \\mbox\{\\boldmath$u$\}\_\{k\}\\end\{bmatrix\}\(26\)Which implies the network model is \(𝑸,𝑺,𝑹\\mbox\{\\boldmath$Q$\},\\mbox\{\\boldmath$S$\},\\mbox\{\\boldmath$R$\}\) dissipative\. ∎

### 2\.3Training algorithm for DissipNet

The algorithm to find the weights of the network to guarantee the constraints in Eq\. \([27](https://arxiv.org/html/2609.27186#S2.E27)\)\.

Step 1: Choose𝑫Dsuch thatL=𝑹\+𝑺𝑫\+𝑫T​𝑺T\+𝑫T​𝑸𝑫≻0L=\\mbox\{\\boldmath$R$\}\+\\mbox\{\\boldmath$S$\}\\mbox\{\\boldmath$D$\}\+\\mbox\{\\boldmath$D$\}^\{T\}\\mbox\{\\boldmath$S$\}^\{T\}\+\\mbox\{\\boldmath$D$\}^\{T\}\\mbox\{\\boldmath$Q$\}\\mbox\{\\boldmath$D$\}\\succ 0\[See Appendices[B\.1](https://arxiv.org/html/2609.27186#A2.SS1)\]

If𝑹1=𝑹\+𝑺𝑫\+𝑫T​𝑺T\+𝑫T​𝑸𝑫≻0\\mbox\{\\boldmath$R$\}\_\{1\}=\\mbox\{\\boldmath$R$\}\+\\mbox\{\\boldmath$S$\}\\mbox\{\\boldmath$D$\}\+\\mbox\{\\boldmath$D$\}^\{T\}\\mbox\{\\boldmath$S$\}^\{T\}\+\\mbox\{\\boldmath$D$\}^\{T\}\\mbox\{\\boldmath$Q$\}\\mbox\{\\boldmath$D$\}\\succ 0, using Schur complement the LMI condition in Eq\. \([22](https://arxiv.org/html/2609.27186#S2.E22)\) can be rewritten as follows

\[𝑷\+𝑪hT​𝑸𝑪h−γ​𝑾h​hT​𝚲−γ​𝚲​𝑾h​h𝚲−γ​𝑾i​h∗−𝑷\]−\[𝑪hT​𝑺T\+𝑪hT​𝑸𝑫−γ​𝚲​𝑾u\]​𝑹1−1​\[𝑪hT​𝑺T\+𝑪hT​𝑸𝑫−γ​𝚲​𝑾u\]T≻0\\begin\{bmatrix\}\\mbox\{\\boldmath$P$\}\+\\mbox\{\\boldmath$C$\}\_\{h\}^\{T\}\\mbox\{\\boldmath$Q$\}\\mbox\{\\boldmath$C$\}\_\{h\}&\-\\gamma\\mbox\{\\boldmath$W$\}\_\{hh\}^\{T\}\\bm\{\\Lambda\}\\\\ \-\\gamma\\bm\{\\Lambda\}\\mbox\{\\boldmath$W$\}\_\{hh\}&\\bm\{\\Lambda\}\-\\gamma\\mbox\{\\boldmath$W$\}^\{\*\}\_\{ih\}\-\\mbox\{\\boldmath$P$\}\\\\ \\end\{bmatrix\}\-\\\\ \\begin\{bmatrix\}\\mbox\{\\boldmath$C$\}\_\{h\}^\{T\}\\mbox\{\\boldmath$S$\}^\{T\}\+\\mbox\{\\boldmath$C$\}\_\{h\}^\{T\}\\mbox\{\\boldmath$Q$\}\\mbox\{\\boldmath$D$\}\\\\ \-\\gamma\\bm\{\\Lambda\}\\mbox\{\\boldmath$W$\}\_\{u\}\\end\{bmatrix\}\\mbox\{\\boldmath$R$\}\_\{1\}^\{\-1\}\\begin\{bmatrix\}\\mbox\{\\boldmath$C$\}\_\{h\}^\{T\}\\mbox\{\\boldmath$S$\}^\{T\}\+\\mbox\{\\boldmath$C$\}\_\{h\}^\{T\}\\mbox\{\\boldmath$Q$\}\\mbox\{\\boldmath$D$\}\\\\ \-\\gamma\\bm\{\\Lambda\}\\mbox\{\\boldmath$W$\}\_\{u\}\\end\{bmatrix\}^\{T\}\\succ 0\(27\)
Step 2\. Choose𝑪h,𝑩,𝚲​𝑾u\\mbox\{\\boldmath$C$\}\_\{h\},\\mbox\{\\boldmath$B$\},\\bm\{\\Lambda\}\\mbox\{\\boldmath$W$\}\_\{u\}as free variables, we will choose𝑷,𝚲\\mbox\{\\boldmath$P$\},\\bm\{\\Lambda\}such that

\[𝑷\+𝑪hT​𝑸𝑪h−γ​𝑾h​hT​𝚲−γ​𝚲​𝑾h​h𝚲−γ​𝑾i​h∗−𝑷\]≻𝑿T​𝑿\+\[𝑪hT​𝑺T\+𝑪hT​𝑸𝑫−γ​𝚲​𝑾u\]​𝑹1−1​\[𝑺𝑪h\+𝑫T​𝑸𝑪h−γ​𝑾uT​𝚲\]\\begin\{bmatrix\}\\mbox\{\\boldmath$P$\}\+\\mbox\{\\boldmath$C$\}\_\{h\}^\{T\}\\mbox\{\\boldmath$Q$\}\\mbox\{\\boldmath$C$\}\_\{h\}&\-\\gamma\\mbox\{\\boldmath$W$\}\_\{hh\}^\{T\}\\bm\{\\Lambda\}\\\\ \-\\gamma\\bm\{\\Lambda\}\\mbox\{\\boldmath$W$\}\_\{hh\}&\\bm\{\\Lambda\}\-\\gamma\\mbox\{\\boldmath$W$\}^\{\*\}\_\{ih\}\-\\mbox\{\\boldmath$P$\}\\\\ \\end\{bmatrix\}\\succ\\mbox\{\\boldmath$X$\}^\{T\}\\mbox\{\\boldmath$X$\}\\\\ \+\\begin\{bmatrix\}\\mbox\{\\boldmath$C$\}\_\{h\}^\{T\}\\mbox\{\\boldmath$S$\}^\{T\}\+\\mbox\{\\boldmath$C$\}\_\{h\}^\{T\}\\mbox\{\\boldmath$Q$\}\\mbox\{\\boldmath$D$\}\\\\ \-\\gamma\\bm\{\\Lambda\}\\mbox\{\\boldmath$W$\}\_\{u\}\\end\{bmatrix\}\\mbox\{\\boldmath$R$\}\_\{1\}^\{\-1\}\[\\mbox\{\\boldmath$S$\}\\mbox\{\\boldmath$C$\}\_\{h\}\+\\mbox\{\\boldmath$D$\}^\{T\}\\mbox\{\\boldmath$Q$\}\\mbox\{\\boldmath$C$\}\_\{h\}\\quad\-\\gamma\\mbox\{\\boldmath$W$\}\_\{u\}^\{T\}\\bm\{\\Lambda\}\]\(28\)Where𝑿Xis a free variable\. Denote

𝑯=\[𝑯11𝑯12𝑯21𝑯22\]=𝑿𝑿T\+\[𝑪hT​𝑺T\+𝑪hT​𝑸𝑫−γ​𝚲​𝑾u\]​𝑹1−1​\[𝑺𝑪h\+𝑫T​𝑸𝑪h−γ​𝑾uT​𝚲\]\\mbox\{\\boldmath$H$\}=\\begin\{bmatrix\}\\mbox\{\\boldmath$H$\}\_\{11\}&\\mbox\{\\boldmath$H$\}\_\{12\}\\\\ \\mbox\{\\boldmath$H$\}\_\{21\}&\\mbox\{\\boldmath$H$\}\_\{22\}\\end\{bmatrix\}=\\mbox\{\\boldmath$X$\}\\mbox\{\\boldmath$X$\}^\{T\}\\\\ \+\\begin\{bmatrix\}\\mbox\{\\boldmath$C$\}\_\{h\}^\{T\}\\mbox\{\\boldmath$S$\}^\{T\}\+\\mbox\{\\boldmath$C$\}\_\{h\}^\{T\}\\mbox\{\\boldmath$Q$\}\\mbox\{\\boldmath$D$\}\\\\ \-\\gamma\\bm\{\\Lambda\}\\mbox\{\\boldmath$W$\}\_\{u\}\\end\{bmatrix\}\\mbox\{\\boldmath$R$\}\_\{1\}^\{\-1\}\[\\mbox\{\\boldmath$S$\}\\mbox\{\\boldmath$C$\}\_\{h\}\+\\mbox\{\\boldmath$D$\}^\{T\}\\mbox\{\\boldmath$Q$\}\\mbox\{\\boldmath$C$\}\_\{h\}\\quad\-\\gamma\\mbox\{\\boldmath$W$\}\_\{u\}^\{T\}\\bm\{\\Lambda\}\]\(29\)We bring another matrix𝑯∗\\mbox\{\\boldmath$H$\}\*that satisfies

𝑯∗=\[𝑯11\+ϵ1​𝑰𝑯12∗\+𝒀𝑯12∗T\+𝒀T𝑯22\+ϵ1​𝑰\]≻\[𝑯11𝑯12𝑯12T𝑯22\]\\mbox\{\\boldmath$H$\}^\{\*\}=\\begin\{bmatrix\}\\mbox\{\\boldmath$H$\}\_\{11\}\+\\epsilon\_\{1\}\\mbox\{\\boldmath$I$\}&\\mbox\{\\boldmath$H$\}\_\{12\}^\{\*\}\+\\mbox\{\\boldmath$Y$\}\\\\ \\mbox\{\\boldmath$H$\}\_\{12\}^\{\*T\}\+\\mbox\{\\boldmath$Y$\}^\{T\}&\\mbox\{\\boldmath$H$\}\_\{22\}\+\\epsilon\_\{1\}\\mbox\{\\boldmath$I$\}\\end\{bmatrix\}\\succ\\begin\{bmatrix\}\\mbox\{\\boldmath$H$\}\_\{11\}&\\mbox\{\\boldmath$H$\}\_\{12\}\\\\ \\mbox\{\\boldmath$H$\}\_\{12\}^\{T\}&\\mbox\{\\boldmath$H$\}\_\{22\}\\end\{bmatrix\}\(30\)where𝑯12∗\\mbox\{\\boldmath$H$\}\_\{12\}^\{\*\}is the matrix that is obtained from𝑯12\\mbox\{\\boldmath$H$\}\_\{12\}by zeroing out all non\-diagonal block matrices while preserving the diagonal block matrices in the same form as matrix𝑾h​h\\mbox\{\\boldmath$W$\}\_\{hh\}in Eq\. \([11](https://arxiv.org/html/2609.27186#S2.E11)\)\.𝒀∈ℝL​q×L​q\\mbox\{\\boldmath$Y$\}\\in\\mathbb\{R\}^\{Lq\\times Lq\}is the trainable matrix that has the same form as𝑾h​h\\mbox\{\\boldmath$W$\}\_\{hh\}\. For Eq\. \([30](https://arxiv.org/html/2609.27186#S2.E30)\) to be satisfied,ϵ1\\epsilon\_\{1\}will be chosen such that

ϵ1≥‖𝑯12∗\+𝒀−𝑯12‖\\epsilon\_\{1\}\\geq\\norm\{\\H\_\{12\}^\* \+ \\Y\-\\H\_\{12\}\}\(31\)
Finally,𝑷Pandγ​𝑾h​hT​𝚲\\gamma\\mbox\{\\boldmath$W$\}\_\{hh\}^\{T\}\\bm\{\\Lambda\}can be chosen as follows

𝑷=𝑯11−𝑪hT​𝑸𝑪h\+ϵ1​𝑰≻0\\mbox\{\\boldmath$P$\}=\\mbox\{\\boldmath$H$\}\_\{11\}\-\\mbox\{\\boldmath$C$\}\_\{h\}^\{T\}\\mbox\{\\boldmath$Q$\}\\mbox\{\\boldmath$C$\}\_\{h\}\+\\epsilon\_\{1\}\\mbox\{\\boldmath$I$\}\\succ 0\(32\)
−γ​𝑾h​hT​𝚲=𝑯12∗\+𝒀\-\\gamma\\mbox\{\\boldmath$W$\}\_\{hh\}^\{T\}\\bm\{\\Lambda\}=\\mbox\{\\boldmath$H$\}\_\{12\}^\{\*\}\+\\mbox\{\\boldmath$Y$\}\(33\)whered​i​a​g​\(⋅\)diag\(\\cdot\)andR​e​L​U​\(⋅\)ReLU\(\\cdot\)are diagonal operator and Rectified Linear Unit \(ReLU\) function, respectively\. It is easily seen that the Eqs\. \([32](https://arxiv.org/html/2609.27186#S2.E32)\-[33](https://arxiv.org/html/2609.27186#S2.E33)\) guarantee that

\[𝑷\+𝑪T​𝑸𝑪−γ​𝑾h​hT​𝚲−γ​𝚲​𝑾h​h𝚲−γ​𝑾i​h∗−𝑷\]≻\[𝑯11\+ϵ1​𝑰𝑯12∗𝑯12∗T𝑯22\+ϵ1​𝑰\]\\begin\{bmatrix\}\\mbox\{\\boldmath$P$\}\+\\mbox\{\\boldmath$C$\}^\{T\}\\mbox\{\\boldmath$Q$\}\\mbox\{\\boldmath$C$\}&\-\\gamma\\mbox\{\\boldmath$W$\}\_\{hh\}^\{T\}\\bm\{\\Lambda\}\\\\ \-\\gamma\\bm\{\\Lambda\}\\mbox\{\\boldmath$W$\}\_\{hh\}&\\bm\{\\Lambda\}\-\\gamma\\mbox\{\\boldmath$W$\}^\{\*\}\_\{ih\}\-\\mbox\{\\boldmath$P$\}\\\\ \\end\{bmatrix\}\\\\ \\succ\\begin\{bmatrix\}\\mbox\{\\boldmath$H$\}\_\{11\}\+\\epsilon\_\{1\}\\mbox\{\\boldmath$I$\}&\\mbox\{\\boldmath$H$\}\_\{12\}^\{\*\}\\\\ \\mbox\{\\boldmath$H$\}\_\{12\}^\{\*T\}&\\mbox\{\\boldmath$H$\}\_\{22\}\+\\epsilon\_\{1\}\\mbox\{\\boldmath$I$\}\\end\{bmatrix\}\(34\)By choosingγ​𝚲​𝑾i​h\\gamma\\bm\{\\Lambda\}\\mbox\{\\boldmath$W$\}\_\{ih\}freely and𝚲\\bm\{\\Lambda\}as in Eq\. \([35](https://arxiv.org/html/2609.27186#S2.E35)\)

𝚲=\(‖𝑯22\+𝑷\+γ​𝑾i​h∗\+ϵ1​𝑰‖\+ϵ2\)​𝑰\\bm\{\\Lambda\}=\\left\(\\norm\{ \\H\_\{22\}\+\\P\+\\gamma\\W^\*\_\{ih\} \+\\epsilon\_1\\I\}\+\\epsilon\_\{2\}\\right\)\\mbox\{\\boldmath$I$\}\(35\)whereϵ2\>0\.\\epsilon\_\{2\}\>0\.We can also obtain that

\[𝑷\+𝑪hT​𝑸𝑪h−γ​𝑾h​hT​𝚲−γ​𝚲​𝑾h​h𝚲−γ​𝑾i​h∗−𝑷\]≻\[𝑯11𝑯12𝑯21𝑯22\]\\begin\{bmatrix\}\\mbox\{\\boldmath$P$\}\+\\mbox\{\\boldmath$C$\}\_\{h\}^\{T\}\\mbox\{\\boldmath$Q$\}\\mbox\{\\boldmath$C$\}\_\{h\}&\-\\gamma\\mbox\{\\boldmath$W$\}\_\{hh\}^\{T\}\\bm\{\\Lambda\}\\\\ \-\\gamma\\bm\{\\Lambda\}\\mbox\{\\boldmath$W$\}\_\{hh\}&\\bm\{\\Lambda\}\-\\gamma\\mbox\{\\boldmath$W$\}^\{\*\}\_\{ih\}\-\\mbox\{\\boldmath$P$\}\\\\ \\end\{bmatrix\}\\succ\\begin\{bmatrix\}\\mbox\{\\boldmath$H$\}\_\{11\}&\\mbox\{\\boldmath$H$\}\_\{12\}\\\\ \\mbox\{\\boldmath$H$\}\_\{21\}&\\mbox\{\\boldmath$H$\}\_\{22\}\\end\{bmatrix\}\(36\)Using Eq\. \([35](https://arxiv.org/html/2609.27186#S2.E35)\), matrices𝑾h​h,𝑾i​h,𝑾u\\mbox\{\\boldmath$W$\}\_\{hh\},\\mbox\{\\boldmath$W$\}\_\{ih\},\\mbox\{\\boldmath$W$\}\_\{u\}then can be calculated, given𝚲​𝑾h​h,𝚲​𝑾i​h,𝚲​𝑾u\\bm\{\\Lambda\}\\mbox\{\\boldmath$W$\}\_\{hh\},\\bm\{\\Lambda\}\\mbox\{\\boldmath$W$\}\_\{ih\},\\bm\{\\Lambda\}\\mbox\{\\boldmath$W$\}\_\{u\}\. From Eq\. \([34](https://arxiv.org/html/2609.27186#S2.E34)\) and Eq\. \([36](https://arxiv.org/html/2609.27186#S2.E36)\), Eq\. \([27](https://arxiv.org/html/2609.27186#S2.E27)\) is satisfied\.

## 3Case studies

In this section, the proposed DissipNet will be evaluated through two robotic modeling examples: a mass\-spring\-damper model and a two degree\-of\-freedom planar manipulator, which were both proven to be dissipative\.

### 3\.1Mass\-spring\-damper system

The equation of the model of a mass\-spring\-damper \(MSD\) model can be expressed as

m​x¨​\(t\)\+b​x˙​\(t\)\+k​x​\(t\)=f⁡\(t\)m\\ddot\{x\}\(t\)\+b\\dot\{x\}\(t\)\+kx\(t\)=f\(t\)\(37\)Herem,b,km,b,kare mass, damping, and spring coefficients, respectively\.ffis the input of the system, andxxis the position of the mass\. The energy relation of the MSD system is as follows

∫0Tf⁡\(t\)​x˙=12​k​x2​\(T\)−12​k​x2​\(0\)\+12​m​x˙2​\(T\)−12​m​x˙2​\(0\)\+∫0Tb​x˙​\(t\)​𝑑t\\int\_\{0\}^\{T\}f\(t\)\\dot\{x\}=\\frac\{1\}\{2\}kx^\{2\}\(T\)\-\\frac\{1\}\{2\}kx^\{2\}\(0\)\+\\\\ \\frac\{1\}\{2\}m\\dot\{x\}^\{2\}\(T\)\-\\frac\{1\}\{2\}m\\dot\{x\}^\{2\}\(0\)\+\\int\_\{0\}^\{T\}b\\dot\{x\}\(t\)dt\(38\)Eq\. \([38](https://arxiv.org/html/2609.27186#S3.E38)\) implies that the continous system is dissipative with the following storage function and the supply rate

V⁡\(x\)=12​k​x2​\(t\)\+12​m​x˙2​\(t\)V\(x\)=\\frac\{1\}\{2\}kx^\{2\}\(t\)\+\\frac\{1\}\{2\}m\\dot\{x\}^\{2\}\(t\)\(39\)w⁡\(u,y\)=\[𝒚T​f\]​\[0000−b120120\]​\[𝒚T​f\]Tw\(u,y\)=\\begin\{bmatrix\}\\mbox\{\\boldmath$y$\}^\{T\}f\\end\{bmatrix\}\\begin\{bmatrix\}0&0&0\\\\ 0&\-b&\\frac\{1\}\{2\}\\\\ 0&\\frac\{1\}\{2\}&0\\end\{bmatrix\}\\begin\{bmatrix\}\\mbox\{\\boldmath$y$\}^\{T\}f\\end\{bmatrix\}^\{T\}\(40\)where𝒚=\[xx˙\]T\\mbox\{\\boldmath$y$\}=\\begin\{bmatrix\}x&\\dot\{x\}\\end\{bmatrix\}^\{T\}

Parameters of the MSD model are follows: m = 1 \(kg\), k =4\.2​N/m4\.2N/m, b =2\.5​N​s/m2\.5Ns/m, and the sampling time was 0\.05s\.

### 3\.2Two DOF manipulator

Modeling of a planar two link manipulator is used to evaluate the performance of the DissipNet\. The equation of motion of the manipulator can be expressed as follows\[[20](https://arxiv.org/html/2609.27186#bib.bib10)\]

𝑴⁡\(𝒒\)​𝒒¨\+𝑪⁡\(𝒒,𝒒˙\)​𝒒˙\+𝑭f​r​i​c​t​i​o​n=𝝉\\mbox\{\\boldmath$M$\}\(\\mbox\{\\boldmath$q$\}\)\\ddot\{\\mbox\{\\boldmath$q$\}\}\+\\mbox\{\\boldmath$C$\}\(\\mbox\{\\boldmath$q$\},\\dot\{\\mbox\{\\boldmath$q$\}\}\)\\dot\{\\mbox\{\\boldmath$q$\}\}\+\\mbox\{\\boldmath$F$\}\_\{friction\}=\\mbox\{\\boldmath$\\tau$\}\(41\)where𝒒=\[q1q2\]T\\mbox\{\\boldmath$q$\}=\[q\_\{1\}\\quad q\_\{2\}\]^\{T\}denotes the joint angles of the manipulator\.

𝑴⁡\(𝒒\)=\[m11m12m21m22\]\\displaystyle\\mbox\{\\boldmath$M$\}\(\\mbox\{\\boldmath$q$\}\)=\\begin\{bmatrix\}m\_\{11\}&m\_\{12\}\\\\ m\_\{21\}&m\_\{22\}\\end\{bmatrix\}\(42\)𝑪⁡\(𝒒,𝒒˙\)=\[c11c12c21c22\]\\displaystyle\\mbox\{\\boldmath$C$\}\(\\mbox\{\\boldmath$q$\},\\dot\{\\mbox\{\\boldmath$q$\}\}\)=\\begin\{bmatrix\}c\_\{11\}&c\_\{12\}\\\\ c\_\{21\}&c\_\{22\}\\end\{bmatrix\}\(43\)𝑭f​r​i​c​t​i​o​n=\[b1​q˙1b2​q˙2\]T\\displaystyle\\mbox\{\\boldmath$F$\}\_\{friction\}=\\begin\{bmatrix\}b\_\{1\}\\dot\{q\}\_\{1\}&b\_\{2\}\\dot\{q\}\_\{2\}\\end\{bmatrix\}^\{T\}\(44\)𝝉=\[τ1τ2\]T\\displaystyle\\mbox\{\\boldmath$\\tau$\}=\\begin\{bmatrix\}\\tau\_\{1\}&\\tau\_\{2\}\\end\{bmatrix\}^\{T\}\(45\)𝑴⁡\(𝒒\),𝑪⁡\(𝒒,𝒒˙\),𝑭f​r​i​c​t​i​o​n,𝝉\\mbox\{\\boldmath$M$\}\(\\mbox\{\\boldmath$q$\}\),\\mbox\{\\boldmath$C$\}\(\\mbox\{\\boldmath$q$\},\\dot\{\\mbox\{\\boldmath$q$\}\}\),\\mbox\{\\boldmath$F$\}\_\{friction\},\\mbox\{\\boldmath$\\tau$\}are the mass matrix, Corriolis and Centrifugal matrix, friction torques, and the vector of applied torques to the joints, respectively\.m11=m1​l124\+m2​\(l12\+l22/4\+l1​l2​c​o​s​\(q2\)\)\+Iz​1\+Iz​2m\_\{11\}=m\_\{1\}\\frac\{l\_\{1\}^\{2\}\}\{4\}\+m\_\{2\}\(l\_\{1\}^\{2\}\+l\_\{2\}^\{2\}/4\+l\_\{1\}l\_\{2\}cos\(q\_\{2\}\)\)\+I\_\{z1\}\+I\_\{z2\}m12=m2​\(14\+l1​l24​c​o​s​\(q2\)\)\+Iz​2m\_\{12\}=m\_\{2\}\(\\frac\{1\}\{4\}\+\\frac\{l\_\{1\}l\_\{2\}\}\{4\}cos\(q\_\{2\}\)\)\+I\_\{z2\},m21=m12,m22=m2​l224\+Iz​2m\_\{21\}=m\_\{12\},m\_\{22\}=m\_\{2\}\\frac\{l\_\{2\}^\{2\}\}\{4\}\+I\_\{z2\}c11=−m2​l1​l2​q˙2​s​i​n​\(q2\),c12=−12​l1​l2​q˙2​s​i​n​\(q2\)c\_\{11\}=\-m\_\{2\}l\_\{1\}l\_\{2\}\\dot\{q\}\_\{2\}sin\(q\_\{2\}\),c\_\{12\}=\-\\frac\{1\}\{2\}l\_\{1\}l\_\{2\}\\dot\{q\}\_\{2\}sin\(q\_\{2\}\)c21=−14​m2​l1​l2​q˙2​s​i​n​\(q2\)\+12​m2​l1​l2​q˙1​s​i​n​\(q2\)c\_\{21\}=\-\\frac\{1\}\{4\}m\_\{2\}l\_\{1\}l\_\{2\}\\dot\{q\}\_\{2\}sin\(q\_\{2\}\)\+\\frac\{1\}\{2\}m\_\{2\}l\_\{1\}l\_\{2\}\\dot\{q\}\_\{1\}sin\(q\_\{2\}\),𝒄22=−14​m2​l1​l2​q˙1​s​i​n​\(q2\)\\mbox\{\\boldmath$c$\}\_\{22\}=\-\\frac\{1\}\{4\}m\_\{2\}l\_\{1\}l\_\{2\}\\dot\{q\}\_\{1\}sin\(q\_\{2\}\)

τ1=τm​1\+g⁡\[l1​\(m12\+m2\)​cos⁡q1\+m2​l22​cos⁡\(q1\+q2\)\]−px​\[l1​sin⁡q1\+l2​sin⁡\(q1\+q2\)\]\+py​\[l1​cos⁡q1\+l2​cos⁡\(q1\+q2\)\],τ2=τm​2\+12​m2​g​l2​cos⁡\(q1\+q2\)−px​l2​sin⁡\(q1\+q2\)\+py​l2​cos⁡\(q1\+q2\)\.\\tau\_\{1\}=\\tau\_\{m1\}\+g\\left\[l\_\{1\}\\left\(\\frac\{m\_\{1\}\}\{2\}\+m\_\{2\}\\right\)\\cos q\_\{1\}\+\\frac\{m\_\{2\}l\_\{2\}\}\{2\}\\cos\(q\_1 \+ q\_2\)\\right\]\\\\ \-p\_\{x\}\\left\[l\_\{1\}\\sin q\_\{1\}\+l\_\{2\}\\sin\(q\_1 \+ q\_2\)\\right\]\+\\\\ p\_\{y\}\\left\[l\_\{1\}\\cos q\_\{1\}\+l\_\{2\}\\cos\(q\_1 \+ q\_2\)\\right\],\\\\ \\tau\_\{2\}=\\tau\_\{m2\}\+\\frac\{1\}\{2\}m\_\{2\}gl\_\{2\}\\cos\(q\_1 \+ q\_2\)\\\\ \-p\_\{x\}l\_\{2\}\\sin\(q\_1 \+ q\_2\)\+p\_\{y\}l\_\{2\}\\cos\(q\_1 \+ q\_2\)\.\(46\)Here,mi​\(i=1,2\)m\_\{i\}\(i=1,2\)denotes the mass ofit​hi\_\{th\}link;lil\_\{i\}is the length ofit​hi\_\{th\}link, andIiI\_\{i\}is the moment of inertia of the link about the axis passing through the center of the mass and perpendicular to the plane of motion\.pxp\_\{x\}andpyp\_\{y\}are the x and y components of the external forces, which are considered to be zero in this paper\.τm​1,τm​2\\tau\_\{m1\},\\tau\_\{m2\}are active joint torques\. The parameters of the 2\-dof manipulator werem1=m2=1​\(k​g\)m\_\{1\}=m\_\{2\}=1\(kg\),l1=l2=1​\(m\)l\_\{1\}=l\_\{2\}=1\(m\),Iz​1=Iz​2=0\.1​k​g​m2I\_\{z1\}=I\_\{z2\}=0\.1kgm^\{2\},b1=b2=0\.01​N​m​sb\_\{1\}=b\_\{2\}=0\.01Nms\. The sampling time was 0\.05s\. It can be shown that the input\-output pair of the continuous system𝒇fand𝒒˙\\dot\{\\mbox\{\\boldmath$q$\}\}is passive\[[5](https://arxiv.org/html/2609.27186#bib.bib17)\]\.

## 4Evaluation results

The DissipNet will be used to model the input\-output relationship of the models described above\. Moreover, we will compare the performance of our model with an RNN model without dissipativity property \(Naive RNN\) and a physics\-informed neural network model \(PINN\)\[[18](https://arxiv.org/html/2609.27186#bib.bib13)\]\.

Here, the idea of the PINN modelling approach is to integrate the system’s known dynamics by adding the physical model into the cost with a regularization coefficient\. In particular, the loss for the MSD and 2\-DOF manipulator will be as in Eq\.[47](https://arxiv.org/html/2609.27186#S4.E47)and Eq\.[48](https://arxiv.org/html/2609.27186#S4.E48), respectively\.

ℒ=∑i=1N\[ℒ⁡\(𝒚i,𝒚id\)\+λ​\(m​xi¨\+b​xi˙\+k​xi−fi\)2\]\\mathcal\{L\}=\\sum\_\{i=1\}^\{N\}\\left\[\\mathcal\{L\}\(\\mbox\{\\boldmath$y$\}\_\{i\},\\mbox\{\\boldmath$y$\}^\{d\}\_\{i\}\)\+\\lambda\(m\\ddot\{x\_\{i\}\}\+b\\dot\{x\_\{i\}\}\+kx\_\{i\}\-f\_\{i\}\)^\{2\}\\right\]\(47\)ℒ=∑i=1N\[ℒ⁡\(𝒚i,𝒚id\)\+λ​𝒓iT​𝒓i\]\\mathcal\{L\}=\\sum\_\{i=1\}^\{N\}\\left\[\\mathcal\{L\}\(\\mbox\{\\boldmath$y$\}\_\{i\},\\mbox\{\\boldmath$y$\}^\{d\}\_\{i\}\)\+\\lambda\\mbox\{\\boldmath$r$\}\_\{i\}^\{T\}\\mbox\{\\boldmath$r$\}\_\{i\}\\right\]\(48\)where𝒓i=𝑴⁡\(𝒒i\)​𝒒i¨\+𝑪⁡\(𝒒i,𝒒i˙\)​𝒒i˙\+𝑭f​r​i​c​t​i​o​n,i−𝝉i\\mbox\{\\boldmath$r$\}\_\{i\}=\\mbox\{\\boldmath$M$\}\(\\mbox\{\\boldmath$q$\}\_\{i\}\)\\ddot\{\\mbox\{\\boldmath$q$\}\_\{i\}\}\+\\mbox\{\\boldmath$C$\}\(\\mbox\{\\boldmath$q$\}\_\{i\},\\dot\{\\mbox\{\\boldmath$q$\}\_\{i\}\}\)\\dot\{\\mbox\{\\boldmath$q$\}\_\{i\}\}\+\\mbox\{\\boldmath$F$\}\_\{friction,i\}\-\\mbox\{\\boldmath$\\tau$\}\_\{i\}

The dissipavitivy of the DissipNet model is evaluated using Eq\. \([3](https://arxiv.org/html/2609.27186#S2.E3)\) whereV=𝝂k​𝑷​𝝂kV=\\bm\{\\nu\}\_\{k\}\\mbox\{\\boldmath$P$\}\\bm\{\\nu\}\_\{k\}as shown in Eq\. \([3](https://arxiv.org/html/2609.27186#S2.E3)\)\. For the PINN model \(continous model\), the condition to show the dissipativity is

∫0Tky​u​𝑑t≥V⁡\(Tk\)−V⁡\(0\)\\int\_\{0\}^\{T\_\{k\}\}yudt\\geq V\(T\_\{k\}\)\-V\(0\)\(49\)All simulations were conducted on a desktop workstation with an Intel\(R\) Core\(TM\) i9\-9900K CPU @ 3\.6 GHz, 128 GB RAM, and an NVIDIA GeForce RTX 2080 Ti GPU\. For both the 2\-DOF manipulator and the MSD systems, the RNN and DRNN models share the same architecture with a hidden dimension of 64, and 2 recurrent layers\. The PINN baseline is a 4\-layer feed\-forward network with a hidden dimension of 128\. All models were trained using the MSE loss and the Adam optimizer with a learning rate of10−310^\{\-3\}for 5000 and 10000 epochs for the 2\-DOF manipulator and MSD systems, respectively\.

![Refer to caption](https://arxiv.org/html/2609.27186v1/fig/compare_msd.jpg)Figure 1:Comparison of DissipNet model \(DRNN\) and PINN model on a mass\-spring\-damper \(MSD\) model under an external input force\. The second and third subplots from the top show that all RNN, DRNN and PINN models can capture the dynamics of the MSD system\. However, the fourth subplot shows that the PINN model does not preserve dissipativity property, as opposed to the dissipativity property of the DNN model implied in the last subplot\. Here, the shaded color shows the area where the value is negative\. Data from 0s to 40s were used for training, and data from 40s to 80s were used for testing\.![Refer to caption](https://arxiv.org/html/2609.27186v1/fig/compare_msd_noise.jpg)Figure 2:Comparison between RNN, DRNN and PINN models when the training data are subjected to noises\. Here, Gaussian noise with zero mean and standard deviationσ=0\.1\\sigma=0\.1is added to the outputs in the training\. It is seen that the DRNN obtained better results compared with RNN and PINN while preserving dissipativity property, which indicates the robustness of the proposed DissipNet modelled when modeling dissipative systems\.![Refer to caption](https://arxiv.org/html/2609.27186v1/fig/compare_2dof.jpg)Figure 3:Comparison of the DissipNet model \(DRNN\), the Naive RNN model \(RNN\) and PINN model on the 2\-dof manipulator modeling\. The second and third subplots from the top show that all the naive RNN model, DRNN and PINN model can capture the dynamics of the system, however, the fourth subplot indicates that the PINN model is not dissipative as opposed to the DRNN model implied from the bottom subplot\. Here, data from 0s to 20s were used for training, and data from 20s to 40s were used for testing\.![Refer to caption](https://arxiv.org/html/2609.27186v1/fig/compare_2dof_noise.jpg)Figure 4:Comparison between RNN, DRNN, and PINN models when the training data are corrupted by noise \(Gaussian noise with zero mean and standard deviationσ=0\.1\\sigma=0\.1\)\. The results show that the proposed DRNN achieves superior prediction performance and exhibits lower sensitivity to noise compared with both RNN and PINN, indicating the robustness of the proposed DissipNet\. Data from0​s0\\,\\mathrm\{s\}to20​s20\\,\\mathrm\{s\}are used for training, while data from20​s20\\,\\mathrm\{s\}to40​s40\\,\\mathrm\{s\}are used for testing\.Fig\.[1](https://arxiv.org/html/2609.27186#S4.F1)compares the proposed DissipNet \(DRNN\) with a naive RNN and a PINN on the MSD model\. While all three networks can capture the system dynamics of the MSD system, notable differences arise in terms of energy behavior\. In particular, the PINN does not preserve the dissipativity property, as evidenced by the corresponding energy evolution\. In contrast, the energy plot of the proposed DRNN \(bottom subplot\) clearly demonstrates dissipative behavior, confirming that the dissipativity constraint is effectively enforced by the network architecture\. Fig\.[2](https://arxiv.org/html/2609.27186#S4.F2)compares the performance of the models when the training data are corrupted by noise\. The results show that the proposed DRNN achieves superior prediction accuracy with less noise sensitivity compared with both the RNN and PINN, while simultaneously preserving the dissipativity property\. Table[1](https://arxiv.org/html/2609.27186#S4.T1)presents a quantitative comparison of the mean prediction errors and corresponding standard deviations obtained using RNN, PINN, and the proposed DRNN for the MSD model, under varying training noise conditions and numbers of training samples\. In all evaluated scenarios, the proposed DRNN consistently achieves significantly lower prediction errors than the standard RNN, demonstrating the effectiveness of enforcing dissipativity constraints in improving learning stability and accuracy\. ForN=200N=200with training noise, DRNN reduces the mean error from6\.53×10−26\.53\\times 10^\{\-2\}to1\.12×10−21\.12\\times 10^\{\-2\}, corresponding to an improvement of over80%80\\%, while also exhibiting a markedly smaller variance\. A similar trend is observed forN=400N=400, where DRNN attains a mean error of1\.24×10−21\.24\\times 10^\{\-2\}compared to6\.59×10−26\.59\\times 10^\{\-2\}for RNN, indicating that the performance advantage of DRNN persists as the dataset size increases\.

When compared with PINN, DRNN demonstrates superior robustness to training noise, particularly in the low\-data regime\. ForN=200N=200with noise, DRNN achieves nearly a fivefold reduction in error relative to PINN, while forN=400N=400the improvement remains substantial\. In the absence of training noise, both DRNN and PINN benefit from increased data availability; however, DRNN consistently attains the lowest error across all cases, reaching6\.0×10−46\.0\\times 10^\{\-4\}forN=400N=400\. These results collectively indicate that incorporating dissipativity directly into the network dissipativity directly into the network architecture leads to improved generalization and noise robustness compared with both purely data\-driven RNNs and physics\-informed PINNs in modeling MSD dynamics\.

Table 1:Comparing mean errors \(±\\pmstd\) of prediction performance using RNN, DRNN and PINN models for the MSD model \(N is the number of training samples\)Table 2:Comparing mean errors \(±\\pmstd\) of prediction performance using RNN, DRNN and PINN models for the 2DOF manipulator model \(N is the number of training samples\)In Fig\.[3](https://arxiv.org/html/2609.27186#S4.F3), the proposed DRNN is compared with a naive RNN and a PINN on the modeling of a 2\-DOF manipulator\. Similar to the MSD system, the PINN fails to preserve the dissipativity property of the modeled system, in contrast to the proposed DRNN\. Fig\.[4](https://arxiv.org/html/2609.27186#S4.F4)illustrates the performance of all models when the robot joint angles are corrupted by Gaussian noise\. It is seen that DRNN is less sensitive to noise, and in both Fig\.[3](https://arxiv.org/html/2609.27186#S4.F3)and Fig\.[4](https://arxiv.org/html/2609.27186#S4.F4), the proposed DRNN consistently achieves lower prediction errors than both the RNN and PINN, while preserving dissipativity\. This superior modeling performance is further confirmed by the quantitative results reported in Table[2](https://arxiv.org/html/2609.27186#S4.T2)\. Table[2](https://arxiv.org/html/2609.27186#S4.T2)compares the prediction error of RNN, PINN, and the proposed DRNN under different data lengths and noise conditions\. Across all settings, DRNN consistently achieves lower mean error and reduced variance compared to the standard RNN, indicating improved accuracy and robustness due to the imposed dissipative structure\. In particular, forN=200N=200, with noise, DRNN reduces the error by approximately22%22\\%relative to RNN, while forN=400N=400the reduction increases to about40%40\\%\. Compared with PINN, DRNN performs competitively under noisy conditions and shows a clear advantage in noise\-free cases, especially as the data length increases, where DRNN attains the lowest error among all methods\. These results suggest that explicitly enforcing dissipativity leads to better generalization and noise robustness than both purely data\-driven RNNs and physics\-regularized PINNs\.

## 5Conclusions

In this work, we proposed a dissipative recurrent network model that can learn the dynamics and preserve the dissipative property of a known dissipative nonlinear model\. This was realized through the design and learning algorithm of the network structure so that the weights satisfy dissipative constraints and they can be solved with non\-constrained optimization method\. The proposed network was evaluated and compared with a PINN model and a naive RNN on a mass\-spring\-damper system and a 2\-dof planar manipulator, which are dissipative with particular input\-output pairs\. The results showed the superior performance of our dissipative RNN network compared to Naive RNN and PINN networks in guaranteeing both good modeling performance and physical meaning preservation\. Moreover, it can be observed that, while PINN networks do not guarantee the preservation of the dissipativity property of the system, they also require prior knowledge of the system dynamics in the form of explicit equations, which is not needed in the proposed network\.

This is expected to provide a useful tool for designing a more effective and physically meaningful network applicable across a wide range of applications\. Research on the modeling of the proposed model on more complex case studies, and improving the training time will be our future work\.

## Appendix APROOF OF EQ\. \([24](https://arxiv.org/html/2609.27186#S2.E24)\)

Left and right multiplying Eq\. \([22](https://arxiv.org/html/2609.27186#S2.E22)\) by\[𝝂kT𝝂k\+1T𝒖kT\]\[\\bm\{\\nu\}\_\{k\}^\{T\}\\quad\\bm\{\\nu\}\_\{k\+1\}^\{T\}\\quad\\mbox\{\\boldmath$u$\}\_\{k\}^\{T\}\]and\[𝝂kT𝝂k\+1T𝒖kT\]T\[\\bm\{\\nu\}\_\{k\}^\{T\}\\quad\\bm\{\\nu\}\_\{k\+1\}^\{T\}\\quad\\mbox\{\\boldmath$u$\}\_\{k\}^\{T\}\]^\{T\}respectively, one can obtain

𝝂kT​𝑷​𝝂k−𝝂k\+1T​𝑷​𝝂k\+1\+𝝂kT​𝑪hT​𝑸𝑪h​𝝂k\+2​𝝂kT​𝑫T​𝑸𝑪h​𝒖k\+2​𝝂kT​𝑪hT​𝑺T​𝒖k\+𝒖kT​𝑹1​𝒖k−2​γ​𝒗kT​𝚲​𝝂k\+1\+𝝂k\+1T​𝚲​𝝂k\+1≥0\\bm\{\\nu\}\_\{k\}^\{T\}\\mbox\{\\boldmath$P$\}\\bm\{\\nu\}\_\{k\}\-\\bm\{\\nu\}\_\{k\+1\}^\{T\}\\mbox\{\\boldmath$P$\}\\bm\{\\nu\}\_\{k\+1\}\+\\bm\{\\nu\}\_\{k\}^\{T\}\\mbox\{\\boldmath$C$\}\_\{h\}^\{T\}\\mbox\{\\boldmath$Q$\}\\mbox\{\\boldmath$C$\}\_\{h\}\\bm\{\\nu\}\_\{k\}\+2\\bm\{\\nu\}\_\{k\}^\{T\}\\mbox\{\\boldmath$D$\}^\{T\}\\mbox\{\\boldmath$Q$\}\\mbox\{\\boldmath$C$\}\_\{h\}\\mbox\{\\boldmath$u$\}\_\{k\}\\\\ \+2\\bm\{\\nu\}\_\{k\}^\{T\}\\mbox\{\\boldmath$C$\}\_\{h\}^\{T\}\\mbox\{\\boldmath$S$\}^\{T\}\\mbox\{\\boldmath$u$\}\_\{k\}\+\\mbox\{\\boldmath$u$\}\_\{k\}^\{T\}\\mbox\{\\boldmath$R$\}\_\{1\}\\mbox\{\\boldmath$u$\}\_\{k\}\-2\\gamma\\mbox\{\\boldmath$v$\}\_\{k\}^\{T\}\\bm\{\\Lambda\}\\bm\{\\nu\}\_\{k\+1\}\+\\bm\{\\nu\}\_\{k\+1\}^\{T\}\\bm\{\\Lambda\}\\bm\{\\nu\}\_\{k\+1\}\\geq 0\(50\)taking into account that𝒗k=𝑾u​𝒖k\+𝑾i​h​𝝂k\+1\+𝑾h​h​𝝂k\\mbox\{\\boldmath$v$\}\_\{k\}=\\mbox\{\\boldmath$W$\}\_\{u\}\\mbox\{\\boldmath$u$\}\_\{k\}\+\\mbox\{\\boldmath$W$\}\_\{ih\}\\bm\{\\nu\}\_\{k\+1\}\+\\mbox\{\\boldmath$W$\}\_\{hh\}\\bm\{\\nu\}\_\{k\}, andVk\+1​\(𝝂k\+1\)−Vk​\(𝝂k\)=𝝂k\+1T​𝑷​𝝂k\+1−𝝂kT​𝑷​𝝂kV\_\{k\+1\}\(\\bm\{\\nu\}\_\{k\+1\}\)\-V\_\{k\}\(\\bm\{\\nu\}\_\{k\}\)=\\bm\{\\nu\}\_\{k\+1\}^\{T\}\\mbox\{\\boldmath$P$\}\\bm\{\\nu\}\_\{k\+1\}\-\\bm\{\\nu\}\_\{k\}^\{T\}\\mbox\{\\boldmath$P$\}\\bm\{\\nu\}\_\{k\}, and

\[𝒚kT𝒖kT\]​\[𝑸𝑺T𝑺𝑹\]​\[𝒚k𝒖k\]=𝝂kT​𝑪hT​𝑸𝑪h​𝝂k\+2​𝝂kT​𝑫T​𝑸𝑪h​𝒖k\+2​𝝂kT​𝑪hT​𝑺T​𝒖k\+𝒖kT​𝑹1​𝒖k\[\\mbox\{\\boldmath$y$\}\_\{k\}^\{T\}\\quad\\mbox\{\\boldmath$u$\}\_\{k\}^\{T\}\]\\begin\{bmatrix\}\\mbox\{\\boldmath$Q$\}&\\mbox\{\\boldmath$S$\}^\{T\}\\\\ \\mbox\{\\boldmath$S$\}&\\mbox\{\\boldmath$R$\}\\end\{bmatrix\}\\begin\{bmatrix\}\\mbox\{\\boldmath$y$\}\_\{k\}\\\\ \\mbox\{\\boldmath$u$\}\_\{k\}\\end\{bmatrix\}=\\bm\{\\nu\}\_\{k\}^\{T\}\\mbox\{\\boldmath$C$\}\_\{h\}^\{T\}\\mbox\{\\boldmath$Q$\}\\mbox\{\\boldmath$C$\}\_\{h\}\\bm\{\\nu\}\_\{k\}\\\\ \+2\\bm\{\\nu\}\_\{k\}^\{T\}\\mbox\{\\boldmath$D$\}^\{T\}\\mbox\{\\boldmath$Q$\}\\mbox\{\\boldmath$C$\}\_\{h\}\\mbox\{\\boldmath$u$\}\_\{k\}\+2\\bm\{\\nu\}\_\{k\}^\{T\}\\mbox\{\\boldmath$C$\}\_\{h\}^\{T\}\\mbox\{\\boldmath$S$\}^\{T\}\\mbox\{\\boldmath$u$\}\_\{k\}\+\\mbox\{\\boldmath$u$\}\_\{k\}^\{T\}\\mbox\{\\boldmath$R$\}\_\{1\}\\mbox\{\\boldmath$u$\}\_\{k\}\(51\)where𝑹1=𝑹\+𝑺𝑫\+𝑫T​𝑺T\+𝑫T​𝑸𝑫\\mbox\{\\boldmath$R$\}\_\{1\}=\\mbox\{\\boldmath$R$\}\+\\mbox\{\\boldmath$S$\}\\mbox\{\\boldmath$D$\}\+\\mbox\{\\boldmath$D$\}^\{T\}\\mbox\{\\boldmath$S$\}^\{T\}\+\\mbox\{\\boldmath$D$\}^\{T\}\\mbox\{\\boldmath$Q$\}\\mbox\{\\boldmath$D$\}\.

Eq\. \([50](https://arxiv.org/html/2609.27186#A1.E50)\) then can be re\-written as

Vk\+1​\(𝝂k\+1\)−Vk​\(𝝂k\)≤\[𝒚kT𝒖kT\]​\[𝑸𝑺T𝑺𝑹\]​\[𝒚k𝒖k\]\+\[𝒗kT𝝂k\+1T\]​\[𝟎−γ​𝚲−γ​𝚲𝚲\]​\[𝒗k𝝂k\+1\]V\_\{k\+1\}\(\\bm\{\\nu\}\_\{k\+1\}\)\-V\_\{k\}\(\\bm\{\\nu\}\_\{k\}\)\\leq\[\\mbox\{\\boldmath$y$\}\_\{k\}^\{T\}\\quad\\mbox\{\\boldmath$u$\}\_\{k\}^\{T\}\]\\begin\{bmatrix\}\\mbox\{\\boldmath$Q$\}&\\mbox\{\\boldmath$S$\}^\{T\}\\\\ \\mbox\{\\boldmath$S$\}&\\mbox\{\\boldmath$R$\}\\end\{bmatrix\}\\begin\{bmatrix\}\\mbox\{\\boldmath$y$\}\_\{k\}\\\\ \\mbox\{\\boldmath$u$\}\_\{k\}\\end\{bmatrix\}\\\\ \+\[\\mbox\{\\boldmath$v$\}\_\{k\}^\{T\}\\quad\\bm\{\\nu\}\_\{k\+1\}^\{T\}\]\\begin\{bmatrix\}\\mbox\{\\boldmath$0$\}&\-\\gamma\\bm\{\\Lambda\}\\\\ \-\\gamma\\bm\{\\Lambda\}&\\bm\{\\Lambda\}\\end\{bmatrix\}\\begin\{bmatrix\}\\mbox\{\\boldmath$v$\}\_\{k\}\\\\ \\bm\{\\nu\}\_\{k\+1\}\\end\{bmatrix\}\(52\)

## Appendix BPROOF OF EQ\. \([25](https://arxiv.org/html/2609.27186#S2.E25)\)

The slopes of widely\-used activation functionsϕ⁡\(\)\\phi\(\), such as ReLU, tanh, exponential linear functions, are restricted by the range \[alpha, beta\]\. We have

α​𝒗k≤𝝂k\+1=ϕ⁡\(𝒗k\)−ϕ⁡\(𝟎\)≤β​𝝂k\\alpha\\mbox\{\\boldmath$v$\}\_\{k\}\\leq\\bm\{\\nu\}\_\{k\+1\}=\\phi\(\\mbox\{\\boldmath$v$\}\_\{k\}\)\-\\phi\(\\mbox\{\\boldmath$0$\}\)\\leq\\beta\\bm\{\\nu\}\_\{k\}\(53\)Therefore

\(𝝂k\+1−α​𝒗k\)​𝚲​\(𝝂k\+1−β​𝒗k\)\(\\bm\{\\nu\}\_\{k\+1\}\-\\alpha\\mbox\{\\boldmath$v$\}\_\{k\}\)\\bm\{\\Lambda\}\(\\bm\{\\nu\}\_\{k\+1\}\-\\beta\\mbox\{\\boldmath$v$\}\_\{k\}\)\(54\)
\[𝒗kT𝝂k\+1T\]​\[ρ​𝚲−γ​𝚲−γ​𝚲𝚲\]​\[𝒗k𝝂k\+1\]≤0\[\\mbox\{\\boldmath$v$\}\_\{k\}^\{T\}\\quad\\bm\{\\nu\}\_\{k\+1\}^\{T\}\]\\begin\{bmatrix\}\\rho\\bm\{\\Lambda\}&\-\\gamma\\bm\{\\Lambda\}\\\\ \-\\gamma\\bm\{\\Lambda\}&\\bm\{\\Lambda\}\\end\{bmatrix\}\\begin\{bmatrix\}\\mbox\{\\boldmath$v$\}\_\{k\}\\\\ \\bm\{\\nu\}\_\{k\+1\}\\end\{bmatrix\}\\leq 0\(55\)where,ρ=α​β,γ=12​\(α\+β\)\\rho=\\alpha\\beta,\\gamma=\\frac\{1\}\{2\}\(\\alpha\+\\beta\)\. From assumption[1](https://arxiv.org/html/2609.27186#Thmassumption1), Eq\. \([55](https://arxiv.org/html/2609.27186#A2.E55)\) becomes

\[𝒗kT𝝂k\+1T\]​\[𝟎−γ​𝚲−γ​𝚲𝚲\]​\[𝒗k𝝂k\+1\]≤0\[\\mbox\{\\boldmath$v$\}\_\{k\}^\{T\}\\quad\\bm\{\\nu\}\_\{k\+1\}^\{T\}\]\\begin\{bmatrix\}\\mbox\{\\boldmath$0$\}&\-\\gamma\\bm\{\\Lambda\}\\\\ \-\\gamma\\bm\{\\Lambda\}&\\bm\{\\Lambda\}\\end\{bmatrix\}\\begin\{bmatrix\}\\mbox\{\\boldmath$v$\}\_\{k\}\\\\ \\bm\{\\nu\}\_\{k\+1\}\\end\{bmatrix\}\\leq 0\(56\)
### B\.1Choose𝑫Dsuch that

𝑹1=𝑹\+𝑺𝑫\+𝑫T​𝑺T\+𝑫T​𝑸𝑫≻0\\mbox\{\\boldmath$R$\}\_\{1\}=\\mbox\{\\boldmath$R$\}\+\\mbox\{\\boldmath$S$\}\\mbox\{\\boldmath$D$\}\+\\mbox\{\\boldmath$D$\}^\{T\}\\mbox\{\\boldmath$S$\}^\{T\}\+\\mbox\{\\boldmath$D$\}^\{T\}\\mbox\{\\boldmath$Q$\}\\mbox\{\\boldmath$D$\}\\succ 0\(57\)It is noticed that Eq\. \([57](https://arxiv.org/html/2609.27186#A2.E57)\) will be satisfied if

𝑳=𝑹\+𝑺𝑫\+𝑫T​𝑺T\+𝑫T​𝑸1​𝑫≻0\\mbox\{\\boldmath$L$\}=\\mbox\{\\boldmath$R$\}\+\\mbox\{\\boldmath$S$\}\\mbox\{\\boldmath$D$\}\+\\mbox\{\\boldmath$D$\}^\{T\}\\mbox\{\\boldmath$S$\}^\{T\}\+\\mbox\{\\boldmath$D$\}^\{T\}\\mbox\{\\boldmath$Q$\}\_\{1\}\\mbox\{\\boldmath$D$\}\\succ 0\(58\)where𝑸1=𝑸−ϵ​𝑰\\mbox\{\\boldmath$Q$\}\_\{1\}=\\mbox\{\\boldmath$Q$\}\-\\epsilon\\mbox\{\\boldmath$I$\}with𝑰∈ℝp×p\\mbox\{\\boldmath$I$\}\\in\\mathbb\{R\}^\{p\\times p\}, andϵ\\epsilonis a sufficiently small constant to guarantee that𝑸1≺0\\mbox\{\\boldmath$Q$\}\_\{1\}\\prec 0\.ϵ\\epsiloncan be chosen as 0 in case𝑸≺𝟎\\mbox\{\\boldmath$Q$\}\\prec\\mbox\{\\boldmath$0$\}\. Then𝑸1\\mbox\{\\boldmath$Q$\}\_\{1\}can be represented as𝑸1=−𝑳qT​𝑳q\\mbox\{\\boldmath$Q$\}\_\{1\}=\-\\mbox\{\\boldmath$L$\}\_\{q\}^\{T\}\\mbox\{\\boldmath$L$\}\_\{q\}\.

𝑫Dcan be chosen as follow\[[21](https://arxiv.org/html/2609.27186#bib.bib8)\]

𝑫=−𝑸1−1​𝑺T\+𝑳q−1​𝑵𝑳r\\mbox\{\\boldmath$D$\}=\-\\mbox\{\\boldmath$Q$\}\_\{1\}^\{\-1\}\\mbox\{\\boldmath$S$\}^\{T\}\+\\mbox\{\\boldmath$L$\}\_\{q\}^\{\-1\}\\mbox\{\\boldmath$N$\}\\mbox\{\\boldmath$L$\}\_\{r\}\(59\)where𝑳r∈ℝm×m\\mbox\{\\boldmath$L$\}\_\{r\}\\in\\mathbb\{R\}^\{m\\times m\}is found from Eq\. \([60](https://arxiv.org/html/2609.27186#A2.E60)\)

𝑹−𝑺𝑸1​𝑺T=𝑳rT​𝑳r\\mbox\{\\boldmath$R$\}\-\\mbox\{\\boldmath$S$\}\\mbox\{\\boldmath$Q$\}\_\{1\}\\mbox\{\\boldmath$S$\}^\{T\}=\\mbox\{\\boldmath$L$\}\_\{r\}^\{T\}\\mbox\{\\boldmath$L$\}\_\{r\}\(60\)𝑵Nand𝑴Mcan be chosen as follows

ifp≥mp\\geq m

𝑴=𝒀1T​𝒀1\+𝒀2−𝒀2T\+𝒀3T​𝒀3\+ϵ3​𝑰m\\displaystyle\\mbox\{\\boldmath$M$\}=\\mbox\{\\boldmath$Y$\}\_\{1\}^\{T\}\\mbox\{\\boldmath$Y$\}\_\{1\}\+\\mbox\{\\boldmath$Y$\}\_\{2\}\-\\mbox\{\\boldmath$Y$\}\_\{2\}^\{T\}\+\\mbox\{\\boldmath$Y$\}\_\{3\}^\{T\}\\mbox\{\\boldmath$Y$\}\_\{3\}\+\\epsilon\_\{3\}\\mbox\{\\boldmath$I$\}\_\{m\}\(61\)𝑵=\[\(𝑰m−𝑴\)​\(𝑰m\+𝑴\)−1−2​𝒀3​\(𝑰m\+𝑴\)−1\]\\displaystyle\\mbox\{\\boldmath$N$\}=\\begin\{bmatrix\}\(\\mbox\{\\boldmath$I$\}\_\{m\}\-\\mbox\{\\boldmath$M$\}\)\(\\mbox\{\\boldmath$I$\}\_\{m\}\+\\mbox\{\\boldmath$M$\}\)^\{\-1\}\\\\ \-2\\mbox\{\\boldmath$Y$\}\_\{3\}\(\\mbox\{\\boldmath$I$\}\_\{m\}\+\\mbox\{\\boldmath$M$\}\)^\{\-1\}\\end\{bmatrix\}\(62\)where𝒀1∈ℝm×m,𝒀2∈ℝm×m\\mbox\{\\boldmath$Y$\}\_\{1\}\\in\\mathbb\{R\}^\{m\\times m\},\\mbox\{\\boldmath$Y$\}\_\{2\}\\in\\mathbb\{R\}^\{m\\times m\}, and𝒀3∈ℝ\(p−m\)×m\\mbox\{\\boldmath$Y$\}\_\{3\}\\in\\mathbb\{R\}^\{\(p\-m\)\\times m\}are free variables,ϵ3\>0\\epsilon\_\{3\}\>0,𝑰m\\mbox\{\\boldmath$I$\}\_\{m\}is the identity matrix with sizem×mm\\times m\.

ifp<mp<m

𝑴=𝒀1T​𝒀1\+𝒀2−𝒀2T\+𝒀3T​𝒀3\+ϵ3​𝑰p\\displaystyle\\mbox\{\\boldmath$M$\}=\\mbox\{\\boldmath$Y$\}\_\{1\}^\{T\}\\mbox\{\\boldmath$Y$\}\_\{1\}\+\\mbox\{\\boldmath$Y$\}\_\{2\}\-\\mbox\{\\boldmath$Y$\}\_\{2\}^\{T\}\+\\mbox\{\\boldmath$Y$\}\_\{3\}^\{T\}\\mbox\{\\boldmath$Y$\}\_\{3\}\+\\epsilon\_\{3\}\\mbox\{\\boldmath$I$\}\_\{p\}\(63\)𝑵=\[\(𝑰m\+𝑴\)−1​\(𝑰m−𝑴\)−2​\(𝑰m\+𝑴\)−1​𝒀3T\]\\displaystyle\\mbox\{\\boldmath$N$\}=\\begin\{bmatrix\}\(\\mbox\{\\boldmath$I$\}\_\{m\}\+\\mbox\{\\boldmath$M$\}\)^\{\-1\}\(\\mbox\{\\boldmath$I$\}\_\{m\}\-\\mbox\{\\boldmath$M$\}\)\\quad\-2\(\\mbox\{\\boldmath$I$\}\_\{m\}\+\\mbox\{\\boldmath$M$\}\)^\{\-1\}\\mbox\{\\boldmath$Y$\}\_\{3\}^\{T\}\\end\{bmatrix\}\(64\)where𝒀1∈ℝp×p,𝒀2∈ℝp×p\\mbox\{\\boldmath$Y$\}\_\{1\}\\in\\mathbb\{R\}^\{p\\times p\},\\mbox\{\\boldmath$Y$\}\_\{2\}\\in\\mathbb\{R\}^\{p\\times p\}, and𝒀3∈ℝ\(m−p\)×p\\mbox\{\\boldmath$Y$\}\_\{3\}\\in\\mathbb\{R\}^\{\(m\-p\)\\times p\}are free variables,𝑰p\\mbox\{\\boldmath$I$\}\_\{p\}is the identity matrix with sizep×pp\\times p\.

It can be proven that𝑵𝑵T≺𝑰\\mbox\{\\boldmath$N$\}\\mbox\{\\boldmath$N$\}^\{T\}\\prec\\mbox\{\\boldmath$I$\}and𝑹1=𝑳rT​\(𝑰−𝑵𝑵T\)​𝑳r≻0\\mbox\{\\boldmath$R$\}\_\{1\}=\\mbox\{\\boldmath$L$\}\_\{r\}^\{T\}\(\\mbox\{\\boldmath$I$\}\-\\mbox\{\\boldmath$N$\}\\mbox\{\\boldmath$N$\}^\{T\}\)\\mbox\{\\boldmath$L$\}\_\{r\}\\succ 0\. where𝑰Ithe identity matrix with the same size as𝑵𝑵T\\mbox\{\\boldmath$N$\}\\mbox\{\\boldmath$N$\}^\{T\}\.

## Acknowledgment

This material is based upon work supported by the Air Force Office of Scientific Research under award number FA2386\-24\-1\-4039\.

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