Relaxation approximations and bounded variation estimates for some partial differential equations.
We consider control problems governed by semilinear parabolic equations with pointwise state constraints and controls in an Lp-space (p < ∞). We construct a correct relaxed problem, prove some relaxation results, and derive necessary optimality conditions.
We consider control problems governed by semilinear parabolic equations with pointwise state constraints and controls in an -space (). We construct a correct relaxed problem, prove some relaxation results, and derive necessary optimality conditions.
A class of parabolic initial-boundary value problems is considered, where admissible coefficients are given in certain intervals. We are looking for maximal values of the solution with respect to the set of admissible coefficients. We give the abstract general scheme, proposing how to solve such problems with uncertain data. We formulate a general maximization problem and prove its solvability, provided all fundamental assumptions are fulfilled. We apply the theory to certain Fourier obstacle type...
2000 Mathematics Subject Classification: 35K55, 35K60.We investigate the blow-up of the solutions to a nonlinear parabolic system with Robin boundary conditions and time dependent coefficients. We derive sufficient conditions on the nonlinearities and the initial data in order to obtain explicit lower and upper bounds for the blow up time t*.
We study the local exact controllability problem for the Navier-Stokes equations that describe an incompressible fluid flow in a bounded domain with control distributed in a subdomain . The result that we obtained in this paper is as follows. Suppose that is a given solution of the Navier-Stokes equations. Let be a given initial condition and where is small enough. Then there exists a locally distributed control such that the solution of the Navier-Stokes equations:coincides with...
We study the local exact controllability problem for the Navier-Stokes equations that describe an incompressible fluid flow in a bounded domain Ω with control distributed in a subdomain . The result that we obtained in this paper is as follows. Suppose that is a given solution of the Navier-Stokes equations. Let be a given initial condition and where ε is small enough. Then there exists a locally distributed control such that the solution v(t,x) of the Navier-Stokes equations: coincides...
In this paper we deal with the null controllability problem for the heat equation with a memory term by means of boundary controls. For each positive final time T and when the control is acting on the whole boundary, we prove that there exists a set of initial conditions such that the null controllability property fails.
Under natural regularity assumptions on the data the powers of regular elliptic boundary value problems (e.b.v.p.) are shown to be higher order regular e.b.v.p.. This result is used in description of the domains of fractional powers of elliptic operators which information is in order important in regularity considerations for solutions of semilinear parabolic equations. Presented approach allows to avoid C∞-smoothness assumption on the data that is typical in many references.