We prove the existence of solutions to nonlinear parabolic problems of the following type: ∂b(u)⁄ ∂t + A(u) = f + div(Θ(x;t; u)) in Q, u(x;t) = 0 on ∂Ω × [0; T ], b(u)(t = 0) = b(u0) on Ω, where b : ℝ → ℝ is a strictly increasing function of class C 1 , the term A(u) = −div (a(x, t, u, ∇u)) is an operator of Leray-Lions type which satisfies the classical Leray-Lions assumptions of Musielak type, Θ: Ω × [0; T ] × ℝ → ℝ is a Carathéodory, noncoercive function which satisfies the following condition: sup |s|≤k |Θ(·, ·, s)| ∈ Eψ(Q) for all k > 0, where ψ is the Musielak complementary function of Θ, and the second term f belongs to L 1 (Q).
Let $\Omega $ be a bounded domain in ${\mathbb{R}}^n$ with a smooth boundary $\Gamma $. In this work we study the existence of solutions for the following boundary value problem: \[ \frac{\partial ^2 y}{\partial t^2}-M\biggl (\int _\Omega |\nabla y|^2\mathrm{d}x\biggr ) \Delta y -\frac{\partial }{\partial t}\Delta y=f(y) \quad \text{in} Q=\Omega \times (0,\infty ),.1 y=0 \quad \text{in} \Sigma _1=\Gamma _{\!1} \times (0,\infty ), M\biggl (\int _\Omega |\nabla y|^2\mathrm{d}x\biggr ) \frac{\partial y}{\partial \nu } +\frac{\partial }{\partial t}\Bigl (\frac{\partial y}{\partial \nu }\Bigr )=g \quad \text{in} \Sigma _0=\Gamma _{\!0} \times (0,\infty ), y(0)=y_0,\quad \frac{\partial y}{\partial t}\,(0)=y_1 \quad \text{in} \quad \Omega , \qquad \mathrm{(1)}\] where $M$ is a $C^1$-function such that $M(\lambda ) \ge \lambda _0 >0$ for every $\lambda \ge 0$ and $f(y)=|y|^\alpha y$ for $\alpha \ge 0$.
In this paper we consider the optimal control of both operators and parameters for uncertain systems. For the optimal control and identification problem, we show existence of an optimal solution and present necessary conditions of optimality.