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Relaxation of optimal control problems in Lp-SPACES

Nadir Arada (2010)

ESAIM: Control, Optimisation and Calculus of Variations

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.

Relaxation of optimal control problems in 𝖫 𝗉 -spaces

Nadir Arada (2001)

ESAIM: Control, Optimisation and Calculus of Variations

We consider control problems governed by semilinear parabolic equations with pointwise state constraints and controls in an L p -space ( p &lt; ). We construct a correct relaxed problem, prove some relaxation results, and derive necessary optimality conditions.

Reliable solution of parabolic obstacle problems with respect to uncertain data

Ján Lovíšek (2003)

Applications of Mathematics

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...

Remarks on exact controllability for the Navier-Stokes equations

Oleg Yu. Imanuvilov (2001)

ESAIM: Control, Optimisation and Calculus of Variations

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 ω Ω n , n { 2 , 3 } . The result that we obtained in this paper is as follows. Suppose that v ^ ( t , x ) is a given solution of the Navier-Stokes equations. Let v 0 ( x ) be a given initial condition and v ^ ( 0 , · ) - v 0 &lt; ε where ε is small enough. Then there exists a locally distributed control u , supp u ( 0 , T ) × ω such that the solution v ( t , x ) of the Navier-Stokes equations: t v - Δ v + ( v , ) v = p + u + f , div v = 0 , v | Ω = 0 , v | t = 0 = v 0 coincides with...

Remarks on exact controllability for the Navier-Stokes equations

Oleg Yu. Imanuvilov (2010)

ESAIM: Control, Optimisation and Calculus of Variations

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 ω Ω n , n { 2 , 3 } . The result that we obtained in this paper is as follows. Suppose that v ^ ( t , x ) is a given solution of the Navier-Stokes equations. Let v 0 ( x ) be a given initial condition and v ^ ( 0 , · ) - v 0 < ε where ε is small enough. Then there exists a locally distributed control u , supp u ( 0 , T ) × ω such that the solution v(t,x) of the Navier-Stokes equations: t v - Δ v + ( v , ) v = p + u + f , div v = 0 , v | Ω = 0 , v | t = 0 = v 0 coincides...

Remarks on non controllability of the heat equation with memory

Sergio Guerrero, Oleg Yurievich Imanuvilov (2013)

ESAIM: Control, Optimisation and Calculus of Variations

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.

Remarks on the powers of elliptic operators.

Jan W. Cholewa, Tomasz Dlotko (2000)

Revista Matemática Complutense

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.

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