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Inverse Optimal Cardano-Lyapunov Feedback for PDEs with Convection

Mohamed Camil Belhadjoudja, Miroslav Krstic, Mohamed Maghenem, Emmanuel Witrant

TL;DR

The paper tackles inverse optimal control for PDEs with boundary actuation where the CLF derivative is not affine in the control, focusing on depressed cubic and quadratic structures. It derives two inverse-optimal controllers: a Cardano-Lyapunov controller $v=\kappa_c^{*}$ that uniquely minimizes the cost $\mathcal{J}_c$ for cubic dynamics, and a pair of quadratic-root controllers $v=\kappa_q^{*}$ and $v=\beta - \kappa_q^{*}$ that minimize $\mathcal{J}_q}$, with a switching strategy $\\kappa_s^{*}$ to reduce input effort. The proofs establish stabilization and optimality through explicit Lyapunov derivative calculations and a discriminant-based argument for the cubic case, and via squared-term identities for the quadratic case. The results extend inverse-optimal controller design to non-affine PDE settings and offer practical switching schemes to lower actuation effort in convection-diffusion problems.

Abstract

We consider the problem of inverse optimal control design for systems that are not affine in the control. In particular, we consider some classes of partial differential equations (PDEs) with quadratic convection and counter-convection, for which the L2 norm is a control Lyapunov function (CLF) whose derivative has either a depressed cubic or a quadratic dependence in the boundary control input. We also consider diffusive PDEs with or without linear convection, for which a weighted L2 norm is a CLF whose derivative has a quadratic dependence in the control input. For each structure on the derivative of the CLF, we achieve inverse optimality with respect to a meaningful cost functional. For the case where the derivative of the CLF has a depressed cubic dependence in the control, we construct a cost functional for which the unique minimizer is the unique real root of a cubic polynomial: the Cardano-Lyapunov controller. When the derivative of the CLF is quadratic in the control, we construct a cost functional that is minimized by two distinct feedback laws, that correspond to the two distinct real roots of a quadratic equation. We show how to switch from one root to the other to reduce the control effort.

Inverse Optimal Cardano-Lyapunov Feedback for PDEs with Convection

TL;DR

The paper tackles inverse optimal control for PDEs with boundary actuation where the CLF derivative is not affine in the control, focusing on depressed cubic and quadratic structures. It derives two inverse-optimal controllers: a Cardano-Lyapunov controller that uniquely minimizes the cost for cubic dynamics, and a pair of quadratic-root controllers and that minimize , with a switching strategy to reduce input effort. The proofs establish stabilization and optimality through explicit Lyapunov derivative calculations and a discriminant-based argument for the cubic case, and via squared-term identities for the quadratic case. The results extend inverse-optimal controller design to non-affine PDE settings and offer practical switching schemes to lower actuation effort in convection-diffusion problems.

Abstract

We consider the problem of inverse optimal control design for systems that are not affine in the control. In particular, we consider some classes of partial differential equations (PDEs) with quadratic convection and counter-convection, for which the L2 norm is a control Lyapunov function (CLF) whose derivative has either a depressed cubic or a quadratic dependence in the boundary control input. We also consider diffusive PDEs with or without linear convection, for which a weighted L2 norm is a CLF whose derivative has a quadratic dependence in the control input. For each structure on the derivative of the CLF, we achieve inverse optimality with respect to a meaningful cost functional. For the case where the derivative of the CLF has a depressed cubic dependence in the control, we construct a cost functional for which the unique minimizer is the unique real root of a cubic polynomial: the Cardano-Lyapunov controller. When the derivative of the CLF is quadratic in the control, we construct a cost functional that is minimized by two distinct feedback laws, that correspond to the two distinct real roots of a quadratic equation. We show how to switch from one root to the other to reduce the control effort.
Paper Structure (13 sections, 2 theorems, 55 equations)

This paper contains 13 sections, 2 theorems, 55 equations.

Table of Contents

  1. Introduction and Main Results
  2. Problem Statement and Prior Work
  3. Main Results
  4. Examples
  5. Proof of the Main Results
  6. Proof of Theorem \ref{['thm1']}
  7. Step 1. The Feedback $\kappa_{c}^*$ is Asymptotically Stabilizing
  8. Step 2. Optimality
  9. Proof of Theorem \ref{['thm2']}
  10. Step 1. The Feedback Laws $\kappa_{q}^{*}$ and $\beta - \kappa_{q}^{*}$ are Asymptotically Stabilizing
  11. Step 2. Optimality
  12. Switching between Inverse Optimal Controllers
  13. Conclusion

Key Result

Theorem 1

Consider a control system (finite or infinite dimensional) with state $u$, and scalar control input $v$. Suppose that the origin $(u,v):=(0,0)$ is an equilibrium point of the system and that we know a CLF $V$ whose derivative along the control system has the depressed cubic structure in depressed_cu where $\kappa_{c}$ is given in feedback, cardano_lyapu1, cardano_lyapu2. Then the feedback law ach

Theorems & Definitions (9)

  • Example 1: PDEs with quadratic convection
  • Example 2: PDEs with counter-convection
  • Theorem 1
  • Theorem 2
  • Remark 1
  • Remark 2
  • Example 3: PDEs with quadratic convection
  • Example 4: PDEs with linear convection
  • Remark 3