Based on the second derivative, this paper directly establishes the coincidence degree theory of a nonlinear periodic Sturm–Liouville (SL) system. As applications, we study the existence of periodic solutions to the S–L system with some special nonlinear functions by applying Mawhin’s continuation theorem. Some examples and simulations are furnished to inspect the correctness and availability of the chief findings.

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Coincidence Theory of a Nonlinear Periodic Sturm-Liouville System and Its Applications

This chapter concerns the optimal control problem for an electromagnetic wave equation with a potential term depending on a real parameter and with missing initial conditions. By using both the average control notion introduced recently by E. Zuazua to control parameter depending systems and the no-regret method introduced for the optimal control of systems with missing data. The relaxation of averaged no-regret control by the averaged low-regret control sequence transforms the problem into a standard optimal control problem. We prove that the problem of average optimal control admits a unique averaged no-regret control that we characterize by means of optimality systems.

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Averaged No-Regret Control for an Electromagnetic Wave Equation Depending upon a Parameter with Incomplete Initial Conditions

This paper develops a Müntz–Legendre wavelet method for solving a fractional optimal control problem with dynamic constraint as a fractional Sturm–Liouville problem. We first derive the necessary optimality conditions as a two‐point boundary value problem by using the calculus of variation and integration by part formula. The operational matrices for Müntz–Legendre scaling functions have been obtained, and by utilizing it, the two‐point boundary value problem has been converted into a system of algebraic equations. The L2$$ {L}_2 $$ ‐error estimates in the approximation of operations matrices and in the approximation of unknown variable by the Müntz–Legendre wavelet has been derived. In the last, illustrative examples have been taken to show the applicability of the proposed method.

Müntz–Legendre wavelet method for solving Sturm–Liouville fractional optimal control problem with error estimates

Abstract We consider optimal partial regularity for very weak solutions to a class of nonlinear elliptic systems and obtain the general criterion for a very weak solution to be regular in the neighborhood of a given point. First, by Hodge decomposition and the technique of filling holes, we establish the relation between the very weak solution and the classical weak solution. Furthermore, combining the technique of p -harmonic approximation with the method of Hodge decomposition, we obtain the partial regularity result. In particular, the partial regularity we obtained is optimal. 

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Optimal partial regularity for very weak solutions to a class of nonlinear elliptic systems

Null Controllability of Networks Systems

We are concerned with a space-time fractional parabolic initial-boundary value problem of Sturm Liouville type in a general star graph with mixed Dirichlet and Neumann boundary controls. We first give several existence, uniqueness and regularity results of weak and very-weak solutions. Using the notion of no-regret control introduced by Lions, we prove the existence, uniqueness, and characterize the low regret control of a quadratic boundary optimal control problem, then we prove that this low regret control converges to the no-regret control and we provide the associated optimality systems and conditions that characterize that no-regret control.

No-regret and low-regret controls of space-time fractional parabolic Sturm-Liouville equations in a star graph

In the present paper we deal with parabolic fractional initial-boundary value problems of Sturm Liouville type in an interval and in a general star graph. We first give several existence, uniqueness and regularity results of weak and very-weak solutions. We prove the existence and uniqueness of solutions to a quadratic boundary optimal control problem and provide a characterization of the optimal contol via the Euler Lagrange first order optimality conditions. We then investigate the analogous problems for a fractional Sturm Liouville problem in a general star graph with mixed Dirichlet and Neumann boundary controls. The existence and uniqueness of minimizers, and the characterization of the first order optimality conditions are obtained in a general star graph by using the method of Lagrange multipliers.

Optimal control problems of parabolic fractional sturm liouville equations in a star graph

In the present paper, we are concerned with a fractional wave equation of Sturm–Liouville type in a general star graph. We first give several existence, uniqueness and regularity results of weak solutions for the one-dimensional case using the spectral theory; we prove the existence and uniqueness of solutions to a quadratic boundary optimal control problem and provide a characterization of the optimal control via the Euler–Lagrange first-order optimality conditions. We then investigate the analogous problems for a fractional Sturm–Liouville problem in a general star graph with mixed Dirichlet and Neumann boundary conditions and controls of the velocity. We show the existence and uniqueness of minimizers, and by using the first-order optimality conditions with the Lagrange multipliers, we are able to characterize the optimal controls. 

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M. Moutamal

Papers

No-regret and low-regret controls of space-time fractional parabolic Sturm-Liouville equations in a star graph

Optimal control problems of parabolic fractional sturm liouville equations in a star graph

Optimal control of fractional Sturm–Liouville wave equations on a star graph

Quasi-boundary method for a fractional ill-posed problem