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

Abstract

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.

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

“…Based on this paper, we will further continue to study the dynamics of Equation (1) under pulse, delay and random effects. In addition, inspired by the papers [13][14][15][16][17][18][19][20][22][23][24][42][43][44][45][46][47][48][49][50][51][52], we will also study the Sturm-Liouville equation involving fractional differential as well as reaction-diffusion terms in the future.…”

“…The first involves the theoretical and numerical methods and applications of inverse S-L problems (see [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16]). The second focuses on the investigation of some generalized S-L equations, such as the fractional differential S-L equation (see [13][14][15][17][18][19][20][21][22][23][24]) and the S-L equation on time scales (see [25][26][27][28][29][30][31][32][33][34]). The third deals with the S-L problems with certain singularity (see [35][36][37][38]) or discontinuity (see [11,12]).…”

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

“…Only a few works have been done in the area of SLFOCP; for instance, Mophou et al [20] have considered a SLFOCP with quadratic cost and proved the existence of its solutions. Moutamal and Joseph [21] have proved the uniqueness and existence of Optimal control of fractional Sturm-Liouville wave equations on a star graph with quadratic cost. To the best of our knowledge, there is no numerical method has been proposed for the SLFOCP.…”