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Mathematical Control & Related Fields

September 2020 , Volume 10 , Issue 3

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Implicit parametrizations and applications in optimization and control
Dan Tiba
2020, 10(3): 455-470 doi: 10.3934/mcrf.2020006 +[Abstract](554) +[HTML](288) +[PDF](496.41KB)
Abstract:

We discuss necessary conditions (with less Lagrange multipliers), perturbations and general algorithms in non convex optimization problems. Optimal control problems with mixed constraints, governed by ordinary differential equations, are also studied in this context. Our treatment is based on a recent approach to implicit systems, constructing parametrizations of the corresponding manifold, via iterated Hamiltonian equations.

Sparse optimal control for the heat equation with mixed control-state constraints
Eduardo Casas and Fredi Tröltzsch
2020, 10(3): 471-491 doi: 10.3934/mcrf.2020007 +[Abstract](550) +[HTML](285) +[PDF](365.2KB)
Abstract:

A problem of sparse optimal control for the heat equation is considered, where pointwise bounds on the control and mixed pointwise control-state constraints are given. A standard quadratic tracking type functional is to be minimized that includes a Tikhonov regularization term and the \begin{document}$ L^1 $\end{document}-norm of the control accounting for the sparsity. Special emphasis is laid on existence and regularity of Lagrange multipliers for the mixed control-state constraints. To this aim, a duality theorem for linear programming problems in Hilbert spaces is proved and applied to the given optimal control problem.

Optimality conditions in variational form for non-linear constrained stochastic control problems
Laurent Pfeiffer
2020, 10(3): 493-526 doi: 10.3934/mcrf.2020008 +[Abstract](524) +[HTML](265) +[PDF](611.07KB)
Abstract:

Optimality conditions in the form of a variational inequality are proved for a class of constrained optimal control problems of stochastic differential equations. The cost function and the inequality constraints are functions of the probability distribution of the state variable at the final time. The analysis uses in an essential manner a convexity property of the set of reachable probability distributions. An augmented Lagrangian method based on the obtained optimality conditions is proposed and analyzed for solving iteratively the problem. At each iteration of the method, a standard stochastic optimal control problem is solved by dynamic programming. Two academical examples are investigated.

State-constrained semilinear elliptic optimization problems with unrestricted sparse controls
Eduardo Casas and Fredi Tröltzsch
2020, 10(3): 527-546 doi: 10.3934/mcrf.2020009 +[Abstract](543) +[HTML](275) +[PDF](379.73KB)
Abstract:

In this paper, we consider optimal control problems associated with semilinear elliptic equation equations, where the states are subject to pointwise constraints but there are no explicit constraints on the controls. A term is included in the cost functional promoting the sparsity of the optimal control. We prove existence of optimal controls and derive first and second order optimality conditions. In addition, we establish some regularity results for the optimal controls and the associated adjoint states and Lagrange multipliers.

Optimal periodic control for scalar dynamics under integral constraint on the input
Térence Bayen, Alain Rapaport and Fatima-Zahra Tani
2020, 10(3): 547-571 doi: 10.3934/mcrf.2020010 +[Abstract](526) +[HTML](265) +[PDF](436.01KB)
Abstract:

This paper studies a periodic optimal control problem governed by a one-dimensional system, linear with respect to the control \begin{document}$ u $\end{document}, under an integral constraint on \begin{document}$ u $\end{document}. We give conditions for which the value of the cost function at steady state with a constant control \begin{document}$ \bar u $\end{document} can be improved by considering periodic control \begin{document}$ u $\end{document} with average value equal to \begin{document}$ \bar u $\end{document}. This leads to the so-called "over-yielding" met in several applications. With the use of the Pontryagin Maximum Principle, we provide the optimal synthesis of periodic strategies under the integral constraint. The results are illustrated on a single population model in order to study the effect of periodic inputs on the utility of the stock of resource.

A convergent hierarchy of non-linear eigenproblems to compute the joint spectral radius of nonnegative matrices
Stéphane Gaubert and Nikolas Stott
2020, 10(3): 573-590 doi: 10.3934/mcrf.2020011 +[Abstract](513) +[HTML](284) +[PDF](357.26KB)
Abstract:

We show that the joint spectral radius of a finite collection of nonnegative matrices can be bounded by the eigenvalue of a non-linear operator. This eigenvalue coincides with the ergodic constant of a risk-sensitive control problem, or of an entropy game, in which the state space consists of all switching sequences of a given length. We show that, by increasing this length, we arrive at a convergent approximation scheme to compute the joint spectral radius. The complexity of this method is exponential in the length of the switching sequences, but it is quite insensitive to the size of the matrices, allowing us to solve very large scale instances (several matrices in dimensions of order 1000 within a minute). An idea of this method is to replace a hierarchy of optimization problems, introduced by Ahmadi, Jungers, Parrilo and Roozbehani, by a hierarchy of nonlinear eigenproblems. To solve the latter eigenproblems, we introduce a projective version of Krasnoselskii-Mann iteration. This method is of independent interest as it applies more generally to the nonlinear eigenproblem for a monotone positively homogeneous map. Here, this method allows for scalability by avoiding the recourse to linear or semidefinite programming techniques.

Optimal control of the linear wave equation by time-depending BV-controls: A semi-smooth Newton approach
Sebastian Engel and Karl Kunisch
2020, 10(3): 591-622 doi: 10.3934/mcrf.2020012 +[Abstract](589) +[HTML](278) +[PDF](725.36KB)
Abstract:

An optimal control problem for the linear wave equation with control cost chosen as the BV semi-norm in time is analyzed. This formulation enhances piecewise constant optimal controls and penalizes the number of jumps. Existence of optimal solutions and necessary optimality conditions are derived. With numerical realisation in mind, the regularization by \begin{document}$ H^1 $\end{document} functionals is investigated, and the asymptotic behavior as this regularization tends to zero is analyzed. For the \begin{document}$ H^1- $\end{document}regularized problems the semi-smooth Newton algorithm can be used to solve the first order optimality conditions with super-linear convergence rate. Examples are constructed which show that the distributional derivative of an optimal control can be a mix of absolutely continuous measures with respect to the Lebesgue measure, a countable linear combination of Dirac measures, and Cantor measures. Numerical results illustrate and support the analytical results.

Controllability of a system of degenerate parabolic equations with non-diagonalizable diffusion matrix
El Mustapha Ait Ben Hassi, Mohamed Fadili and Lahcen Maniar
2020, 10(3): 623-642 doi: 10.3934/mcrf.2020013 +[Abstract](516) +[HTML](242) +[PDF](440.69KB)
Abstract:

In this paper we study the null controllability of some non diagonalizable degenerate parabolic systems of PDEs, we assume that the diffusion, coupling and controls matrices are constant and we characterize the null controllability by an algebraic condition so called Kalman's rank condition.

Lipschitz stability for some coupled degenerate parabolic systems with locally distributed observations of one component
Brahim Allal, Abdelkarim Hajjaj, Lahcen Maniar and Jawad Salhi
2020, 10(3): 643-667 doi: 10.3934/mcrf.2020014 +[Abstract](674) +[HTML](312) +[PDF](442.33KB)
Abstract:

This article presents an inverse source problem for a cascade system of \begin{document}$ n $\end{document} coupled degenerate parabolic equations. In particular, we prove stability and uniqueness results for the inverse problem of determining the source terms by observations in an arbitrary subdomain over a time interval of only one component and data of the \begin{document}$ n $\end{document} components at a fixed positive time \begin{document}$ T' $\end{document} over the whole spatial domain. The proof is based on the application of a Carleman estimate with a single observation acting on a subdomain.

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