Second order analysis for bang-bang control problems of PDEs
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AuthorCasas Rentería, Eduardo
In this paper, we derive some suﬃcient second order optimality conditions for control problems of partial diﬀerential equations (PDEs) when the cost functional does not involve the usual quadratic term for the control or higher nonlinearities for it. Though not always, in this situation the optimal control is typically bang-bang. Two diﬀerent control problems are studied. The second diﬀers from the ﬁrst in the presence of the L1 norm of the control. This term leads to optimal controls that are sparse and usually take only three diﬀerent values (we call them bang-bang-bang controls). Though the proofs are detailed in the case of a semilinear elliptic state equation, the approach can be extended to parabolic control problems. Some hints are provided in the last section to extend the results.
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