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Recent results on well-posedness and optimal control for a class of generalized fractional Cahn–Hilliard systems

Colli Pierluigi, Gilardi Gianni, Sprekels Jurgen

Control and Cybernetics

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2019

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Vol. 48, No. 1

153--197

EN

In this paper, we give an overview of results for Cahn–Hilliard systems involving fractional operators that have recently been established by the authors of this note. We address problems concerning existence, uniqueness, and regularity of the solutions to the system equations, and we study optimal control problems for the systems. The well-posedness results are valid for a wide class of fractional operators of spectral type and for the typical double-well nonlinearities appearing in the Cahn–Hilliard system equations, namely the classical diﬀerentiable, the logarithmic, and the nondiﬀerentiable double obstacle potentials. While this also applies to the existence of optimal controls in the related optimal control problems, the establishment of ﬁrst-order necessary optimality conditions requires imposing much stronger assumptions on the admissible class of fractional operators. One main reason for this is the necessity of deriving suitable diﬀerentiability properties for the associated control-to-state mapping. Nevertheless, it turns out that also in the singular case of logarithmic potentials, the ﬁrst-order necessary optimality conditions can be established under suitable assumptions, and a “deep quench” approximation, based on the results derived for logarithmic nonlinearities makes even the case of double obstacle potentials accessible.

2

Optimal control for a steady state dead oil isotherm problem

Sidi Ammi M. R., Malinowska A. B., Torres D. F. M.

Control and Cybernetics

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2013

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Vol. 42, no. 2

459--470

EN

We study the optimal control of a steady-state dead oil isotherm problem. The problem is described by a system of nonlinear partial differential equations resulting from the traditional modelling of oil engineering within the framework of mechanics of a continuous medium. Existence and regularity results of the optima control are proved, as well as necessary optimality conditions.

3

Optimal control of delay-differential inclusions with multivalued initial conditions in infinite dimensions

Mordukhovich B. S., Wang D., Wang L.

Control and Cybernetics

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2008

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Vol. 37, no 2

393-428

EN

This paper is devoted to the study of a general class of optimal control problems described by delay-differential inclusions with infinite-dimensional state spaces, endpoints constraints, and multivalued initial conditions. To the best of our knowledge, problems of this type have not been considered in the literature, except for some particular cases when either the state space is finite-dimensional or there is no delay in the dynamics. We develop the method of discrete approximations to derive necessary optimality conditions in the extended Euler-Lagrange form by using advanced tools of variational analysis and generalized differentiation in infinite dimensions. This method consists of the three major parts: (a) constructing a well-posed sequence of discrete-time problems that approximate in an appropriate sense the original continuous-time problem of dynamic optimization; (b) deriving necessary optimality conditions for the approximating discrete-time problems by reducing them to infinite-dimensional problems of mathematical programming and employing then generalized differential calculus; (c) passing finally to the limit in the obtained results for discrete approximations to establish necessary conditions for the given optimal solutions to the original problem. This method is fully realized in the delay-differential systems under consideration.

4

Optimal control of semilinear evolution inclusions via discrete approximations

Mordukhovich B. S., Wang D.

Control and Cybernetics

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2005

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Vol. 34, no 3

849-870

EN

This paper studies a Mayer type optimal control problem with general endpoint constraints for semilinear unbounded evolution inclusions in reflexive and separable Banach spaces. First, we construct a sequence of discrete approximations to the original optimal control problem for evolution inclusions and prove that optimal solutions to discrete approximation problems uniformly converge to a given optimal solution for the original continuous-time problem. Then, based on advanced tools of generalized differentiation, we derive necessary optimality conditions for discrete-time problems under fairly general assumptions. Combining these results with recent achievements of variational analysis in infinite-dimensional spaces, we establish new necessary optimality conditions for constrained continuous-time evolution inclusions by passing to the limit from discrete approximations.

5

Optimal control of constrained delay-differential inclusions with multivalued initial conditions

Mordukhovich B. S., Wang L.

Control and Cybernetics

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2003

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Vol. 32, no 3

585-609

EN

This paper studies a general optimal control problem for nonconvex delay-differential inclusions with endpoint constraints. In contrast to previous publications on this topic, we incorporate time-dependent set constraints on the initial interval, which are specific for systems with delays and provide an additional source for optimization. Our variational analysis is based on well-posed discrete approximations of constrained delay-differential inclusions by a family of time-delayed systems with discrete dynamics and perturbed constraints. Using convergence results for discrete approximations and advanced tools of nonsmooth variational analysis, we derive necessary optimality conditions for constrained delay-differential inclusions in both Euler-Lagrange and Hamiltonian forms involving nonconvex generalized differential constructions for nonsmooth functions, sets, and set-valued mappings.

6

Optimal Harvesting of the Nonlinear Population Dynamics

Anita S.

International Journal of Applied Mathematics and Computer Science

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2000

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Vol. 10, no 1

113-122

EN

This paper deals with an optimal harvesting problem for a nonlinear age-dependent population dynamics. The existence and uniqueness of a positive solution for the model considered is demonstrated. The existence of an optimal harvesting effort and the convergence of a certain fractional step scheme are investigated. Necessary optimality conditions for some approximating problems are established.