On the convergence of controls and cost functionals in some optimal control heterogeneous problems when the homogenization process gives rise to some strange terms

A control scheme for the immunisation of susceptibles in the Kermack-McKendrick epidemic model for a closed population is proposed. The bounded control appears linearly in both dynamics and integral cost functionals and any optimal policies are of the “bang-bang” type. The approach uses Dynamic Programming and Pontryagin's Maximum Principle and allows one, for certain values of the cost and removal rates, to apply necessary and sufficient conditions for optimality and show that a one-switch candidate is the optimal control. In the remaining cases we are still able to show that an optimal control, if it exists, has at most one switch.

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Bi-objective optimal control of some PDEs: Nash equilibria and quasi-equilibria

ESAIM Control Optimisation and Calculus of Variations ◽

10.1051/cocv/2021050 ◽

2021 ◽

Author(s):

Enrique Enrique Fernandez Cara <cara@us. es> ◽

Irene Marín-Gayte

Keyword(s):

Optimal Control ◽

Elliptic Equations ◽

Nash Equilibria ◽

Stokes Equations ◽

Finite Element Approximation ◽

Semilinear Elliptic Equations ◽

Navier Stokes ◽

Element Approximation ◽

Semilinear Elliptic ◽

Cost Functionals

This paper deals with the solution of some multi-objective optimal control problems for several PDEs: linear and semilinear elliptic equations and stationary Navier-Stokes systems. Specifically, we look for Nash equilibria associated with standard cost functionals. For linear and semilinear elliptic equations, we prove the existence of equilibria and we deduce related optimality systems. For stationary Navier-Stokes equations, we prove the existence of Nash quasi-equilibria, i.e. solutions to the optimality system. In all cases, we present some iterative algorithms and, in some of them, we establish convergence results. For the existence and characterization of Nash quasi-equilibria in the Navier-Stokes case, we use the formalism of Dubovitskii and Milyutin. In this context, we also present a finite element approximation and we illustrate the techniques with numerical experiments.

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Multivariate Optimal Control with Payoffs Defined by Submanifold Integrals

Symmetry ◽

10.3390/sym11070893 ◽

2019 ◽

Vol 11 (7) ◽

pp. 893 ◽

Cited By ~ 1

Author(s):

Andreea Bejenaru ◽

Constantin Udriste

Keyword(s):

Optimal Control ◽

Formal Analysis ◽

Linear Connection ◽

Riemannian Structure ◽

State Variables ◽

Riemann Curvature Tensor ◽

Local Coordinates ◽

Geometric Elements ◽

Cost Functionals

This paper adapts the multivariate optimal control theory to a Riemannian setting. In this sense, a coherent correspondence between the key elements of a standard optimal control problem and several basic geometric ingredients is created, with the purpose of generating a geometric version of Pontryagin’s maximum principle. More precisely, the local coordinates on a Riemannian manifold play the role of evolution variables (“multitime”), the Riemannian structure, and the corresponding Levi–Civita linear connection become state variables, while the control variables are represented by some objects with the properties of the Riemann curvature tensor field. Moreover, the constraints are provided by the second order partial differential equations describing the dynamics of the Riemannian structure. The shift from formal analysis to optimal Riemannian control takes deeply into account the symmetries (or anti-symmetries) these geometric elements or equations rely on. In addition, various submanifold integral cost functionals are considered as controlled payoffs.

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Discrete linear optimal control systems with quadratic cost functionals

10.1109/cdc.1973.269233 ◽

1973 ◽

Author(s):

K. Mizukami ◽

J. Neyama ◽

I. McCausland

Keyword(s):

Optimal Control ◽

Control Systems ◽

Linear Optimal Control ◽

Quadratic Cost ◽

Cost Functionals

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A certified RB method for PDE-constrained parametric optimization problems

Communications in Applied and Industrial Mathematics ◽

10.2478/caim-2019-0017 ◽

2019 ◽

Vol 10 (1) ◽

pp. 123-152

Author(s):

Andrea Manzoni ◽

Stefano Pagani

Keyword(s):

Optimal Control ◽

Efficient Solution ◽

Parametric Optimization ◽

Optimization Problems ◽

A Posteriori Error Estimates ◽

Optimal Solution ◽

Posteriori Error ◽

Elliptic Pdes ◽

Reduced Basis ◽

Cost Functionals

Abstract We present a certified reduced basis (RB) framework for the efficient solution of PDE-constrained parametric optimization problems. We consider optimization problems (such as optimal control and optimal design) governed by elliptic PDEs and involving possibly non-convex cost functionals, assuming that the control functions are described in terms of a parameter vector. At each optimization step, the high-fidelity approximation of state and adjoint problems is replaced by a certified RB approximation, thus yielding a very efficient solution through an “optimize-then-reduce” approach. We develop a posteriori error estimates for the solutions of state and adjoint problems, the cost functional, its gradient and the optimal solution. We confirm our theoretical results in the case of optimal control/design problems dealing with potential and thermal flows.

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