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1. A Dual to Lyapanov's Second Method for Linear Systems with Multiple Delays and Implementation using SOS

   https://doi.org/10.1109/TAC.2018.2832470  

   Peet, Matthew M. (May 2018, IEEE Transactions on Automatic Control)

   We present a dual form of Lyapunov-Krasovskii functional which allows the problem of controller synthesis for multi-delay systems to be formulated and solved in a convex manner. First, we give a generalized version of the dual stability condition formulated in terms of Lyapunov operators which are positive, self-adjoint and preserve the structure of the state-space. Second, we provide a class of such operators and express the stability conditions as positivity and negativity of quadratic Lyapunov-Krasovskii functional forms. Next, we adapt the SOS methodology to express positivity and negativity of these forms as LMIs, describing a new set of polynomial manipulation tools designed for this purpose. We apply the resulting LMIs to a battery of numerical examples and demonstrate that the stability conditions are not significantly conservative. Finally, we formulate a test for controller synthesis for systems with multiple delays, apply the test to a numerical example, and simulate the resulting closed-loop system. 

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2. Dual Neural Network (DuNN) method for elliptic partial differential equations and systems

   https://doi.org/10.1016/j.cam.2025.116596  

   Liu, Min; Cai, Zhiqiang; Ramani, Karthik (March 2025, Journal of Computational and Applied Mathematics)

   This paper presents the Dual Neural Network (DuNN) method, a physics-driven numerical method designed to solve elliptic partial differential equations and systems using deep neural network functions and a dual formulation. The underlying elliptic problem is formulated as an optimization of the complementary energy functional in terms of the dual variable, where the Dirichlet boundary condition is weakly enforced in the formulation. To accurately evaluate the complementary energy functional, we employ a novel discrete divergence operator. This discrete operator preserves the underlying physics and naturally enforces the Neumann boundary condition without penalization. For problems without reaction term, we propose an outer-inner iterative procedure that gradually enforces the equilibrium equation through a pseudo-time approach. 

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3. A New State-Space Representation for Coupled PDEs and Scalable Lyapunov Stability Analysis in the SOS Framework

   Peet, M. (December 2018, IEEE Conference on Decision & Control, including the Symposium on Adaptive Processes)

   We present a framework for stability analysis of systems of coupled linear Partial-Differential Equations (PDEs). The class of PDE systems considered in this paper includes parabolic, elliptic and hyperbolic systems with Dirichelet, Neuman and mixed boundary conditions. The results in this paper apply to systems with a single spatial variable and assume existence and continuity of solutions except in such cases when existence and continuity can be inferred from existence of a Lyapunov function. Our approach is based on a new concept of state for PDE systems which allows us to express the derivative of the Lyapunov function as a Linear Operator Inequality directly on L2 and allows for any type of suitably well-posed boundary conditions. This approach obviates the need for integration by parts, spacing functions or similar mathematical encumbrances. The resulting algorithms are implemented in Matlab, tested on several motivating examples, and the codes have been posted online. Numerical testing indicates the approach has little or no conservatism for a large class of systems and can analyze systems of up to 20 coupled PDEs. 

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4. Estimator-Based Output-Feedback Stabilization of Linear Multi-Delay Systems using SOS

   Wu, S.; Peet, M. (January 2019, Proceedings of the IEEE Conference on Decision & Control)

   In this paper, we investigate the estimator-based output feedback control problem of multi-delay systems. This work is an extension of recently developed operator-value LMI framework for infinite-dimensional time-delay systems. Based on the optimal convex state feedback controller and generalized Luenberger observer synthesis conditions we already have, the estimator-based output feedback controller is designed to contain the estimates of both the present state and history of the state. An output feedback controller synthesis condition is proposed using SOS method, which is expressed in a set of LMI/SDP constraints. The simulation examples are displayed to demonstrate the effectiveness and advantages of the proposed results. 

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5. Duality-Based Nested Controller Synthesis from STL Specifications for Stochastic Linear Systems

   https://doi.org/10.1007/978-3-030-00151-3  

   Jha, Susmit; Raj, Sunny; Jha, Sumit Kumar; Shankar, Natarajan (January 2018, 16th International Conference on Formal Modeling and Analysis of Timed Systems, FORMATS 2018)

   We propose an automatic synthesis technique to generate provably correct controllers of stochastic linear dynamical systems for Signal Temporal Logic (STL) specifications. While formal synthesis problems can be directly formulated as exists-forall constraints, the quantifier alternation restricts the scalability of such an approach. We use the duality between a system and its proof of correctness to partially alleviate this challenge. We decompose the controller synthesis into two subproblems, each addressing orthogonal concerns - stabilization with respect to the noise, and meeting the STL specification. The overall controller is a nested controller comprising of the feedback controller for noise cancellation and an open loop controller for STL satisfaction. The correct-by-construction compositional synthesis of this nested controller relies on using the guarantees of the feedback controller instead of the controller itself. We use a linear feedback controller as the stabilizing controller for linear systems with bounded additive noise and over-approximate its ellipsoid stability guarantee with a polytope. We then use this over-approximation to formulate a mixed-integer linear programming (MILP) problem to synthesize an open-loop controller that satisfies STL specifications. 

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