# American Institute of Mathematical Sciences

March  2018, 8(1): 135-153. doi: 10.3934/mcrf.2018006

## On the switching behavior of sparse optimal controls for the one-dimensional heat equation

 1 Institut für Mathematik, Technische Universität Berlin, D-10623 Berlin, Germany 2 Institut für Mathematik, Universität Würzburg, D-97974 Würzburg, Germany

* Corresponding author: Fredi Tröltzsch

Received  April 2017 Revised  September 2017 Published  January 2018

Fund Project: Daniel Wachsmuth was partially supported by the German Research Foundation DFG under project grant Wa 3626/1-1

An optimal boundary control problem for the one-dimensional heat equation is considered. The objective functional includes a standard quadratic terminal observation, a Tikhonov regularization term with regularization parameter $ν$, and the $L^1$-norm of the control that accounts for sparsity. The switching structure of the optimal control is discussed for $ν ≥ 0$. Under natural assumptions, it is shown that the set of switching points of the optimal control is countable with the final time as only possible accumulation point. The convergence of switching points is investigated for $ν \searrow 0$.

Citation: Fredi Tröltzsch, Daniel Wachsmuth. On the switching behavior of sparse optimal controls for the one-dimensional heat equation. Mathematical Control & Related Fields, 2018, 8 (1) : 135-153. doi: 10.3934/mcrf.2018006
##### References:
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Sachs, Algorithmic Methods in Optimal Control volume 47 of Research Notes in Mathematics, Pitman (Advanced Publishing Program), Boston, Mass. -London, 1981. Google Scholar [16] A. Karafiat, The problem of the number of switches in parabolic equations with control, Ann. Polish Math., 34 (1977), 289-316. Google Scholar [17] K. Kunisch and L. Wang, Bang-bang property of time optimal controls of semilinear parabolic equation, Discrete Contin. Dyn. Syst., 36 (2016), 279-302. Google Scholar [18] J. -L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Translated from the French by S. K. Mitter. Die Grundlehren der mathematischen Wissenschaften, Band 170. Springer-Verlag, New York-Berlin, 1971. Google Scholar [19] U. Mackenroth, Some remarks on the numerical solution of bang-bang type optimal control problems, Numer. Funct. Anal. Optim., 5 (1982/83), 457-484. Google Scholar [20] V. J. Mizel and T. I. Seidman, An abstract bang-bang principle and time-optimal boundary control of the heat equation, SIAM J. Control Optim., 35 (1997), 1204-1216. Google Scholar [21] J. P. Raymond and H. Zidani, Pontryagin's principle for state-constrained control problems governed by parabolic equations with unbounded controls, SIAM J. Control Optim., 36 (1998), 1853-1879. Google Scholar [22] E. Sachs, A parabolic control problem with a boundary condition of the Stefan-Boltzmann type, Z. Angew. Math. Mech., 58 (1978), 443-449. Google Scholar [23] K. Schittkowski, Numerical solution of a time-optimal parabolic boundary value control problem, J. Optim. Theory Appl., 27 (1979), 271-290. Google Scholar [24] E. J. P. Georg Schmidt, The "bang-bang" principle for the time-optimal problem in boundary control of the heat equation, SIAM J. Control Optim., 18 (1980), 101-107. Google Scholar [25] E. J. P. Georg Schmidt, Boundary control for the heat equation with nonlinear boundary condition, J. Differential Equations, 78 (1989), 89-121. Google Scholar [26] M. Seydenschwanz, Convergence results for the discrete regularization of linear-quadratic control problems with bang-bang solutions, Comput. Optim. Appl., 61 (2015), 731-760. Google Scholar [27] G. Stadler, Elliptic optimal control problems with $L^1$-control cost and applications for the placement of control devices, Comput. Optim. Appl., 44 (2009), 159-181. Google Scholar [28] F. Tröltzsch, Semidiscrete finite element approximation of parabolic boundary control problems-convergence of switching points, In Optimal Control of Partial Differential Equations, Ⅱ (Oberwolfach, 1986), Internat. Schriftenreihe Numer. Math. , 78, 219-232, Birkhäuser, Basel, 1987. Google Scholar [29] F. Tröltzsch, Optimal Control of Partial Differential Equations volume 112 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2010. Theory, methods and applications, Translated from the 2005 German original by Jürgen Sprekels. Google Scholar [30] A. N. Tychonov and A. A. Samarski, Partial Differential Equations of Mathematical Physics, Vol. Ⅰ Translated by S. Radding. Holden-Day, Inc., San Francisco, Calif. -London-Amsterdam, 1964. Google Scholar [31] D. Wachsmuth and G. Wachsmuth, Regularization error estimates and discrepancy principle for optimal control problems with inequality constraints, Control Cybernet., 40 (2011), 1125-1158. Google Scholar [32] G. Wachsmuth and D. Wachsmuth, Convergence and regularization results for optimal control problems with sparsity functional, ESAIM Control Optim. Calc. Var., 17 (2011), 858-886. Google Scholar [33] G. Wang and L. Wang, The bang-bang principle of time optimal controls for the heat equation with internal controls, Systems Control Lett., 56 (2007), 709-713. Google Scholar [34] L. Wang and Q. Yan, Bang-bang property of time optimal null controls for some semilinear heat equation, SIAM J. Control Optim., 54 (2016), 2949-2964. Google Scholar [35] N. Weck, Über Existenz, Eindeutigkeit und das "Bang-Bang-Prinzip" bei Kontrollproblemen aus der Wärmeleitung, In Numerische Behandlung von Variations und Steuerungsproblemen (Tagungsband, Sonderforschungsber. 72 "Approximation und Optimierung", Inst. Angew. Math., Univ. Bonn, Bonn, 1974), Bonn. Math. Schriften, 77 (1975), 9-19. Google Scholar

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##### References:
 [1] E. Casas, Pontryagin's principle for state-constrained boundary control problems of semilinear parabolic equations, SIAM J. Control Optim., 35 (1997), 1297-1327. Google Scholar [2] E. Casas, Second order analysis for bang-bang control problems of PDEs, SIAM J. Control Optim., 50 (2012), 2355-2372. Google Scholar [3] E. Casas, R. Herzog and G. Wachsmuth, Optimality conditions and error analysis of semilinear elliptic control problems with $L^1$ cost functional, SIAM J. Optim., 22 (2012), 795-820. Google Scholar [4] E. Casas, C. Ryll and F. Tröltzsch, Sparse optimal control of the Schlögl and FitzHugh-Nagumo systems, Comput. Methods Appl. Math., 13 (2013), 415-442. Google Scholar [5] E. Casas, C. Ryll and F. Tröltzsch, Second order and stability analysis for optimal sparse control of the FitzHugh-Nagumo equation, SIAM J. Control Optim., 53 (2015), 2168-2202. Google Scholar [6] V. Dhamo and F. Tröltzsch, Some aspects of reachability for parabolic boundary control problems with control constraints, Comput. Optim. Appl., 50 (2011), 75-110. Google Scholar [7] K. Eppler and F. Tröltzsch, On switching points of optimal controls for coercive parabolic boundary control problems, Optimization, 17 (1986), 93-101. Google Scholar [8] H. O. Fattorini, Time-optimal control of solutions of operational differential equations, SIAM J. on Control, 2 (1964), 54-59. Google Scholar [9] H. O. Fattorini, The time-optimal problem for boundary control of the heat equation, In Calculus of Variations and Control Theory (Proc. Sympos., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1975; Dedicated to Laurence Chisholm Young on the Occasion of His 70th Birthday), Math. Res. Center, Univ. Wisconsin, Publ., Academic Press, New York, 36 (1976), 305-320. Google Scholar [10] H. O. Fattorini, Infinite Dimensional Linear Control Systems volume 201 of North-Holland Mathematics Studies, Elsevier Science B. V., Amsterdam, 2005. Google Scholar [11] K. Glashoff, Restricted approximation by strongly sign-regular kernels: the finite bang-bang principle, J. Approx. Theory, 29 (1980), 212-217. Google Scholar [12] K. Glashoff and W. Krabs, Dualität und Bang-Bang-Prinzip bei einem parabolischen Rand-Kontrollproblem, In Numerische Behandlung von Variations und Steuerungsproblemen (Tagung, Sonderforschungsber. 72 "Approximation und Optimierung", Inst. Angew. Math., Univ. Bonn, Bonn, 1974). Bonn. Math. Schriften, 77 (1975), 1-8. Google Scholar [13] K. Glashoff and E. Sachs, On theoretical and numerical aspects of the bang-bang-principle, Numer. Math., 29 (1977/78), 93-113. Google Scholar [14] K. Glashoff and N. Weck, Boundary control of parabolic differential equations in arbitrary dimensions: supremum-norm problems, SIAM J. Control Optimization, 14 (1976), 662-681. Google Scholar [15] W. A. Gruver and E. Sachs, Algorithmic Methods in Optimal Control volume 47 of Research Notes in Mathematics, Pitman (Advanced Publishing Program), Boston, Mass. -London, 1981. Google Scholar [16] A. Karafiat, The problem of the number of switches in parabolic equations with control, Ann. Polish Math., 34 (1977), 289-316. Google Scholar [17] K. Kunisch and L. Wang, Bang-bang property of time optimal controls of semilinear parabolic equation, Discrete Contin. Dyn. Syst., 36 (2016), 279-302. Google Scholar [18] J. -L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Translated from the French by S. K. Mitter. Die Grundlehren der mathematischen Wissenschaften, Band 170. Springer-Verlag, New York-Berlin, 1971. Google Scholar [19] U. Mackenroth, Some remarks on the numerical solution of bang-bang type optimal control problems, Numer. Funct. Anal. Optim., 5 (1982/83), 457-484. Google Scholar [20] V. J. Mizel and T. I. Seidman, An abstract bang-bang principle and time-optimal boundary control of the heat equation, SIAM J. Control Optim., 35 (1997), 1204-1216. Google Scholar [21] J. P. Raymond and H. Zidani, Pontryagin's principle for state-constrained control problems governed by parabolic equations with unbounded controls, SIAM J. Control Optim., 36 (1998), 1853-1879. Google Scholar [22] E. Sachs, A parabolic control problem with a boundary condition of the Stefan-Boltzmann type, Z. Angew. Math. Mech., 58 (1978), 443-449. Google Scholar [23] K. Schittkowski, Numerical solution of a time-optimal parabolic boundary value control problem, J. Optim. Theory Appl., 27 (1979), 271-290. Google Scholar [24] E. J. P. Georg Schmidt, The "bang-bang" principle for the time-optimal problem in boundary control of the heat equation, SIAM J. Control Optim., 18 (1980), 101-107. Google Scholar [25] E. J. P. Georg Schmidt, Boundary control for the heat equation with nonlinear boundary condition, J. Differential Equations, 78 (1989), 89-121. Google Scholar [26] M. Seydenschwanz, Convergence results for the discrete regularization of linear-quadratic control problems with bang-bang solutions, Comput. Optim. Appl., 61 (2015), 731-760. Google Scholar [27] G. Stadler, Elliptic optimal control problems with $L^1$-control cost and applications for the placement of control devices, Comput. Optim. Appl., 44 (2009), 159-181. Google Scholar [28] F. Tröltzsch, Semidiscrete finite element approximation of parabolic boundary control problems-convergence of switching points, In Optimal Control of Partial Differential Equations, Ⅱ (Oberwolfach, 1986), Internat. Schriftenreihe Numer. Math. , 78, 219-232, Birkhäuser, Basel, 1987. Google Scholar [29] F. Tröltzsch, Optimal Control of Partial Differential Equations volume 112 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2010. Theory, methods and applications, Translated from the 2005 German original by Jürgen Sprekels. Google Scholar [30] A. N. Tychonov and A. A. Samarski, Partial Differential Equations of Mathematical Physics, Vol. Ⅰ Translated by S. Radding. Holden-Day, Inc., San Francisco, Calif. -London-Amsterdam, 1964. Google Scholar [31] D. Wachsmuth and G. Wachsmuth, Regularization error estimates and discrepancy principle for optimal control problems with inequality constraints, Control Cybernet., 40 (2011), 1125-1158. Google Scholar [32] G. Wachsmuth and D. Wachsmuth, Convergence and regularization results for optimal control problems with sparsity functional, ESAIM Control Optim. Calc. Var., 17 (2011), 858-886. Google Scholar [33] G. Wang and L. Wang, The bang-bang principle of time optimal controls for the heat equation with internal controls, Systems Control Lett., 56 (2007), 709-713. Google Scholar [34] L. Wang and Q. Yan, Bang-bang property of time optimal null controls for some semilinear heat equation, SIAM J. Control Optim., 54 (2016), 2949-2964. Google Scholar [35] N. Weck, Über Existenz, Eindeutigkeit und das "Bang-Bang-Prinzip" bei Kontrollproblemen aus der Wärmeleitung, In Numerische Behandlung von Variations und Steuerungsproblemen (Tagungsband, Sonderforschungsber. 72 "Approximation und Optimierung", Inst. Angew. Math., Univ. Bonn, Bonn, 1974), Bonn. Math. Schriften, 77 (1975), 9-19. Google Scholar
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