AbstractWe study an optimal control problem for a quasilinear parabolic equation which has delays in the highest order spatial derivative terms. The cost functional is Lagrange type and some terminal state constraints are presented. A Pontryagin-type maximum principle is derived.
Abstract
The solvability of the nonlocal boundary value problem
𝑢𝑡 = 𝑎(𝑡, 𝑥, 𝑢, 𝑢𝑥)𝑢𝑥𝑥 + 𝑏(𝑡, 𝑥, 𝑢, 𝑢𝑥), 0 ≤ 𝑡 ≤ 𝑇, |𝑥| ≤ 𝑙, 𝑢(0, 𝑥) = 0, 𝑢(𝑡, –𝑙) = 𝑢(𝑡, 𝑙), 𝑢𝑥(𝑡, –𝑙) = 𝑢𝑥(𝑡, 𝑙)
in a class of functions is investigated for a quasilinear parabolic equation. The solution uniqueness follows from the maximum principle.
We investigate an initial-boundary value problem for a quasilinear parabolic equation with inner absorption and nonlinear Neumann boundary condition. We establish, respectively, the conditions on nonlinearity to guarantee thatu(x,t)exists globally or blows up at some finite timet*. Moreover, an upper bound fort*is derived. Under somewhat more restrictive conditions, a lower bound fort*is also obtained.
Optimal control for quasilinear retarded parabolic systems
