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Optimal control of Sobolev type linear equations. (English. Russian original) Zbl 1095.49004
The paper concerns minimization of the cost functional $$J(u)=\Vert Cx(\tau)-z_0(\tau)\Vert^2+\sum_0^{p+1}\int_0^\tau \langle N_qu^{(q)}(t),u^{(q)}(t)\rangle\,dt$$ subject to the Cauchy system $L\dot{x}(t)=Mx(t)+y(t)+Bu(t)$, $x(0)=x_0$. Here the linear, continuous operator $L$ and the linear, closed, densely defined operator $M$ generate a strongly continuous semigroup for the homogeneous equation $L\dot{x}(t)=Mx(t)$ as well as a certain nilpotent operator of degree $p$, at most [see {\it V. E. Fedorov}, “Degenerate strongly continuous semigroup of operators”, St. Petersbg. Math. J. 12, No. 3, 471--489 (2001); translation from Algebra Anal. 12, No. 3, 173--200 (2001; Zbl 0988.47027)], whereas the control $u$ belongs to a Sobolev-Bochner space $W_2^{p+1}$. The main results establish existence and uniqueness of a strong solution $x$ to the Cauchy system, and existence of an optimal control $u$ in case of certain control sets (possibly depending on $y$ and $x_0$). The results are applied to a class of initial-boundary value problems for partial differential equations.
49J20Optimal control problems with PDE (existence)
49K15Optimal control problems with ODE (optimality conditions)
47D06One-parameter semigroups and linear evolution equations
46E35Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
Full Text: DOI
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