Spike Controls for Elliptic and Parabolic PDE

We analyze the use of measures of minimal norm to control elliptic and parabolic equations. We prove the sparsity of the optimal control. In the parabolic case, we prove that the solution of the optimization problem is a Borel measure supported in a set of null Lebesgue measure. In both cases, the approximate controllability can be achieved efficiently by means of controls that are activated in some finite number of pointwise locations. We also analyze the corresponding dual problem.

[1]  Jean-Pierre Raymond,et al.  Pontryagin's Principle for State-Constrained Control Problems Governed by Parabolic Equations with Unbounded Controls , 1998 .

[2]  E. Zuazua,et al.  Approximate Controllability for the Semilinear Heat Equation Involving Gradient Terms , 1999 .

[3]  Carlos E. Kenig,et al.  The Inhomogeneous Dirichlet Problem in Lipschitz Domains , 1995 .

[4]  Jacques-Louis Lions,et al.  Remarks on approximate controllability , 1992 .

[5]  Karl Kunisch,et al.  Approximation of Elliptic Control Problems in Measure Spaces with Sparse Solutions , 2012, SIAM J. Control. Optim..

[6]  Enrique Zuazua,et al.  Controllability and Observability of Partial Differential Equations: Some Results and Open Problems , 2007 .

[7]  Georg Stadler,et al.  Elliptic optimal control problems with L1-control cost and applications for the placement of control devices , 2009, Comput. Optim. Appl..

[8]  W. Rudin.  Real and complex analysis , 1968 .

[9]  E. Casas.  Pontryagin's Principle for State-Constrained Boundary Control Problems of Semilinear Parabolic Equations , 1997 .

[10]  Axel Osses,et al.  On the controllability of the Laplace equation observed on an interior curve , 1998 .

[11]  Roland Herzog,et al.  Directional Sparsity in Optimal Control of Partial Differential Equations , 2012, SIAM J. Control. Optim..

[12]  K. Kunisch,et al.  A duality-based approach to elliptic control problems in non-reflexive Banach spaces , 2011 .

[13]  G. M.,et al.  Partial Differential Equations I , 2023, Applied Mathematical Sciences.

[14]  I. Ekeland,et al.  Convex analysis and variational problems , 1976 .

[15]  Günter Leugering,et al.  L∞-Norm minimal control of the wave equation: on the weakness of the bang-bang principle , 2008 .

[16]  E. Casas.  Control of an elliptic problem with pointwise state constraints , 1986 .

[17]  Gerd Wachsmuth,et al.  Convergence and regularization results for optimal control problems with sparsity functional , 2011 .