An optimal control model for a system of degenerate parabolic integro-differential equations

Abstract

An initial-boundary-value problem for a system of degenerate parabolic integro-differential equations is considered. The sufficient conditions for the existence and uniqueness of its generalized solution and for the existence of at least one optimal control for a given performance functional are obtained. A stable numerical solution to the initial-boundary-value problem is derived for a locally one-dimensional case and conditions are formulated for constructing a stable numerical algorithm of the optimal control problem on a class of piecewise-smooth control functions.

Boundary optimal control for systems described by parabolic problem without initial conditions (in Ukrainian)

“…Вище вказану функцiю p коротко називатимемо розв'язком задачi (11), (12), (14). В наступному пунктi доведено коректнiсть цiєї задачi i отримано вiдповiднi оцiнки її розв'язку (див.…”

“…При вказаних вище умовах iснує єдиний розв'язок u задачi (10) (12), (14) при v = u (тобто, функцiя, що описує спряжений стан, вiдповiдний оптимальному керуванню u), причому в умовi (14) y(x, 0, u), x ∈ Ω, значення при t = 0 розв'язку задачi (5), (6), (8), коли v = u.…”

“…Далi використаємо задачу на спряжений стан при оптимальному керуваннi u, тобто задачу (11), (12), (14) при v = u i для скорочення записiв писатимемо p або p(x, t), (x, t) ∈ Q, замiсть p(u) чи p(x, t; u), (x, t) ∈ Q.…”

Optimal control problem for systems governed by nonlinear parabolic equations without initial conditions

“…In [1,21,24,25] the state of controlled system is described by linear parabolic equations and systems, while in [1] and [21] control functions appears as coefficients at lower derivatives, and in [24,25] the control functions are coefficients at higher derivatives. In [21] the existence and uniqueness of optimal control in the case of final observation was shown and a necessary optimality condition in the form of the generalized rule of Lagrange multipliers was obtained.…”

“…In [21] the existence and uniqueness of optimal control in the case of final observation was shown and a necessary optimality condition in the form of the generalized rule of Lagrange multipliers was obtained. In paper [1] authors proved the existence of at least one optimal control for system governed by a system of general parabolic equations with degenerate discontinuous parabolicity coefficient. In papers [24,25] the authors consider cost function in general form, and as special case it includes different kinds of specific practical optimization problems.…”