No-Regret Optimal Control Characterization for an Ill-Posed Wave Equation

In this paper, we investigate the problem of optimal control for an ill-posed wave equation without using the extra hypothesis of Slater i.e. the set of admissible controls has a non-empty interior. Firstly, by a controllability approach, we make the ill-posed wave equation a well-posed equation with some incomplete data initial condition. The missing data requires us to use the no-regret control notion introduced by Lions to control distributed systems with ncomplete data. After approximating the no-regret control by a low-regret control sequence, we characterize the optimal control by a singular optimality system.

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Two Inverse Problems Solution by Feedback Tracking Control

Axioms ◽

10.3390/axioms10030137 ◽

2021 ◽

Vol 10 (3) ◽

pp. 137

Author(s):

Vladimir Turetsky

Keyword(s):

Optimal Control ◽

Optimal Control Problem ◽

Control Problem ◽

Feedback Linearization ◽

Linear Quadratic ◽

Linear Quadratic Optimal Control ◽

Linear Quadratic Tracking ◽

Ill Posed ◽

Quadratic Optimal Control ◽

Quadratic Tracking

Two inverse ill-posed problems are considered. The first problem is an input restoration of a linear system. The second one is a restoration of time-dependent coefficients of a linear ordinary differential equation. Both problems are reformulated as auxiliary optimal control problems with regularizing cost functional. For the coefficients restoration problem, two control models are proposed. In the first model, the control coefficients are approximated by the output and the estimates of its derivatives. This model yields an approximating linear-quadratic optimal control problem having a known explicit solution. The derivatives are also obtained as auxiliary linear-quadratic tracking controls. The second control model is accurate and leads to a bilinear-quadratic optimal control problem. The latter is tackled in two ways: by an iterative procedure and by a feedback linearization. Simulation results show that a bilinear model provides more accurate coefficients estimates.

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Optimal Control and Additive Perturbations Help in Estimating Ill-Posed and Uncertain Dynamical Systems

Journal of the American Statistical Association ◽

10.1080/01621459.2017.1319841 ◽

2018 ◽

Vol 113 (523) ◽

pp. 1195-1209 ◽

Cited By ~ 3

Author(s):

Quentin Clairon ◽

Nicolas J.-B. Brunel

Keyword(s):

Optimal Control ◽

Dynamical Systems ◽

Uncertain Dynamical Systems ◽

Ill Posed

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Inverse source problems for time-fractional mixed parabolic-hyperbolic-type equations

Journal of Inverse and Ill-Posed Problems ◽

10.1515/jiip-2014-0022 ◽

2015 ◽

Vol 23 (4) ◽

Cited By ~ 3

Author(s):

Pengbin Feng ◽

Erkinjon T. Karimov

Keyword(s):

Wave Equation ◽

Mixed Type ◽

Orthonormal System ◽

Type Equation ◽

Hyperbolic Type ◽

Initial Time ◽

Inverse Source Problem ◽

Ill Posed ◽

Inverse Source Problems ◽

Well Posed

AbstractIn the present paper we consider an inverse source problem for a time-fractional mixed parabolic-hyperbolic equation with Caputo derivatives. In the case when the hyperbolic part of the considered mixed-type equation is the wave equation, the uniqueness of the source and the solution are strongly influenced by the initial time and the problem is generally ill-posed. However, when the hyperbolic part is time-fractional, the problem is well-posed if the end time is large. Our method relies on the orthonormal system of eigenfunctions of the operator with respect to the space variables. Finally, we prove uniqueness and stability of certain weak solutions for the problems under consideration.

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