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<p>We study an optimal control problem for semilinear parabolic equations with infinite horizon, pointwise state contraints and two different types of control constraints (pointwise in space and time, and pointwise in time and <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(L^2\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> in space). First we prove first-order necessary conditions, then we provide a second-order sufficient condition for local optimality. The second-order condition is formulated using an extended cone that considers the infinite horizon and the control and state constraints. This condition is sufficient for strict local optimality in the <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(L^2\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation>-sense. Finally, we address the approximation of the infinite horizon control problem by finite horizon problems. We analyze the convergence of these approximations and provide error estimates.</p>

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Infinite Horizon Control Problems for Semilinear Parabolic Equations with Pointwise State Constraints

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  • 出版模式：

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  • 发表日期：

    2026年3月16日

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  • Lorena Bociu,
  • Eduardo Casas

摘要

We study an optimal control problem for semilinear parabolic equations with infinite horizon, pointwise state contraints and two different types of control constraints (pointwise in space and time, and pointwise in time and \(L^2\) L 2 in space). First we prove first-order necessary conditions, then we provide a second-order sufficient condition for local optimality. The second-order condition is formulated using an extended cone that considers the infinite horizon and the control and state constraints. This condition is sufficient for strict local optimality in the \(L^2\) L 2 -sense. Finally, we address the approximation of the infinite horizon control problem by finite horizon problems. We analyze the convergence of these approximations and provide error estimates.