We consider an optimal control problem with indeenite cost for an abstract model which covers in particular hyperbolic and hyperbolic-like systems in a general bounded domain. Necessary and suucient conditions are given for the synthesis of the optimal control, which is given in terms of the Riccati operator arising from a nonstandard Riccati equation. The theory also extends a nite-dimensional...

We study large time asymptotics of small solutions to the Cauchy problem for nonlinear damped wave equations with a critical nonlinearity { ∂2 t u+ ∂tu−∆u+ λu 2 n = 0, x ∈ Rn, t > 0, u(0, x) = εu0 (x) , ∂tu(0, x) = εu1 (x) , x ∈ Rn, where ε > 0, and space dimensions n = 1, 2, 3. Assume that the initial data u0 ∈ H ∩H, u1 ∈ Hδ−1,0 ∩H−1,δ, where δ > n 2 , weighted Sobolev spaces are H = { φ ∈ L; ...

The existence of a random attractor in H1(R3)×L2(R3) is proved for the damped semilinear stochastic wave equation defined on the entire space R 3. The nonlinearity is allowed to have a cubic growth rate which is referred to as the critical exponent. The uniform pullback estimates on the tails of solutions for large space variables are established. The pullback asymptotic compactness of the rand...