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Optimal LQ-feedback control for a class of first-order hyperbolic distributed parameter systems

Ilyasse Aksikas, Joseph J. Winkin, Denis Dochain (2008)

ESAIM: Control, Optimisation and Calculus of Variations

The Linear-Quadratic (LQ) optimal control problem is studied for a class of first-order hyperbolic partial differential equation models by using a nonlinear infinite-dimensional (distributed parameter) Hilbert state-space description. First the dynamical properties of the linearized model around some equilibrium profile are studied. Next the LQ-feedback operator is computed by using the corresponding operator Riccati algebraic equation whose solution is obtained via a related matrix Riccati differential...

Optimal measures for the fundamental gap of Schrödinger operators

Nicolas Varchon (2010)

ESAIM: Control, Optimisation and Calculus of Variations

We study the potential which minimizes the fundamental gap of the Schrödinger operator under the total mass constraint. We consider the relaxed potential and prove a regularity result for the optimal one, we also give a description of it. A consequence of this result is the existence of an optimal potential under L1 constraints.

Optimal observability of the multi-dimensional wave and Schrödinger equations in quantum ergodic domains

Yannick Privat, Emmanuel Trélat, Enrique Zuazua (2016)

Journal of the European Mathematical Society

We consider the wave and Schrödinger equations on a bounded open connected subset Ω of a Riemannian manifold, with Dirichlet, Neumann or Robin boundary conditions whenever its boundary is nonempty. We observe the restriction of the solutions to a measurable subset ω of Ω during a time interval [ 0 , T ] with T > 0 . It is well known that, if the pair ( ω , T ) satisfies the Geometric Control Condition ( ω being an open set), then an observability inequality holds guaranteeing that the total energy of solutions can be...

Optimal position targeting with stochastic linear-quadratic costs

Stefan Ankirchner, Thomas Kruse (2015)

Banach Center Publications

We consider the dynamic control problem of attaining a target position at a finite time T, while minimizing a linear-quadratic cost functional depending on the position and speed. We assume that the coefficients of the linear-quadratic cost functional are stochastic processes adapted to a Brownian filtration. We provide a probabilistic solution in terms of two coupled backward stochastic differential equations possessing a singularity at the terminal time T. We verify optimality of the candidate...

Optimal snapshot location for computing POD basis functions

Karl Kunisch, Stefan Volkwein (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

The construction of reduced order models for dynamical systems using proper orthogonal decomposition (POD) is based on the information contained in so-called snapshots. These provide the spatial distribution of the dynamical system at discrete time instances. This work is devoted to optimizing the choice of these time instances in such a manner that the error between the POD-solution and the trajectory of the dynamical system is minimized. First and second order optimality systems are given. Numerical...

Optimality and sensitivity for semilinear bang-bang type optimal control problems

Ursula Felgenhauer (2004)

International Journal of Applied Mathematics and Computer Science

In optimal control problems with quadratic terminal cost functionals and systems dynamics linear with respect to control, the solution often has a bang-bang character. Our aim is to investigate structural solution stability when the problem data are subject to perturbations. Throughout the paper, we assume that the problem has a (possibly local) optimum such that the control is piecewise constant and almost everywhere takes extremal values. The points of discontinuity are the switching points. In...

Optimality Conditions for a Nonlinear Boundary Value Problem Using Nonsmooth Analysis

Mohamed Akkouchi, Abdellah Bounabat, Manfred Goebel (2003)

Annales mathématiques Blaise Pascal

We study in this paper a Lipschitz control problem associated to a semilinear second order ordinary differential equation with pointwise state constraints. The control acts as a coefficient of the state equation. The nonlinear part of the equation is governed by a Nemytskij operator defined by a Lipschitzian but possibly nonsmooth function. We prove the existence of optimal controls and obtain a necessary optimality conditions looking somehow to the Pontryagin’s maximum principle. These conditions...

Optimality Conditions for D.C. Vector Optimization Problems under D.C. Constraints

Gadhi, N., Metrane, A. (2004)

Serdica Mathematical Journal

2000 Mathematics Subject Classification: Primary 90C29; Secondary 49K30.In this paper, we establish necessary optimality conditions and sufficient optimality conditions for D.C. vector optimization problems under D.C. constraints. Under additional conditions, some results of [9] and [15] are also recovered.

Optimality conditions for nonconvex variational problems relaxed in terms of Young measures

Tomáš Roubíček (1998)

Kybernetika

The scalar nonconvex variational problems of the minimum-energy type on Sobolev spaces are studied. As the Euler–Lagrange equation dramatically looses selectivity when extended in terms of the Young measures, the correct optimality conditions are sought by means of the convex compactification theory. It turns out that these conditions basically combine one part from the Euler–Lagrange equation with one part from the Weierstrass condition.

Optimality conditions for semilinear parabolic equations with controls in leading term

Hongwei Lou (2011)

ESAIM: Control, Optimisation and Calculus of Variations

An optimal control problem for semilinear parabolic partial differential equations is considered. The control variable appears in the leading term of the equation. Necessary conditions for optimal controls are established by the method of homogenizing spike variation. Results for problems with state constraints are also stated.

Optimality conditions for semilinear parabolic equations with controls in leading term*

Hongwei Lou (2011)

ESAIM: Control, Optimisation and Calculus of Variations

An optimal control problem for semilinear parabolic partial differential equations is considered. The control variable appears in the leading term of the equation. Necessary conditions for optimal controls are established by the method of homogenizing spike variation. Results for problems with state constraints are also stated.

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