Abstract For Ω bounded in ℝm, Q :=(O,T) × Ω, S:=(O,T) × ∂Ω, let ϕ in L2(S) control the evolution of the state u(t) from initial state ω by $$ut + \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{A} u = f \underline {in} Q, \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\beta } {\text{u}} {\text{ = }} \varphi \underline {on} S(*)$$ where $$\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{A}$$ is an elliptic operator (e.g., $$\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{A} = - \Delta$$ ) and $$\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\beta }$$ is a suit-able boundary operator. Call ϕ=ϕ* optimal if it minimizes the quadratic cost functional $$J(\varphi ) : = \int_S {\varphi ^2 + \lambda \int_Q {|u - \hat u} |^2 + \mu \int_\Omega {|u(T) - \hat \omega |^2 .} }$$ With a view to obtaining eventual error estimates for numerical computations, we investigate he regularity of ϕ*. Under suitable hypotheses it is shown, e.g., that ϕ* is analytic from (0,T) to HS(∂Ω) with norm 0(1/|log t| t2/2 − 3/4) as t → 0+. Similar results are obtained for a problem governed by a quasilinear equation: f=f(u) in (*).
Regularity of optimal boundary controls for parabolic equations
1980-01-01
15 pages
Article/Chapter (Book)
Electronic Resource
English
Differential stability of solutions to boundary optimal control problems for parabolic systems
Springer Verlag | 1986
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