Switching controls for analytic semigroups and applications to parabolic systems. (English) Zbl 1466.93018
Summary: In this work, we extend the analysis of the problem of switching controls proposed in
[E. Zuazua, J. Eur. Math. Soc. (JEMS) 13, No. 1, 85–117 (2011; Zbl 1203.49011)].
The problem asks the following question: Assuming that one can control a system using two or more actuators, does there exist a control strategy such that at all times, only one actuator is active? We answer positively when the controlled system corresponds to an analytic semigroup spanned by a positive self-adjoint operator which is null-controllable in arbitrary small times. Similarly to
[loc. cit.], our proof relies on analyticity arguments and will also work in finite dimensional settings and under some further spectral assumptions when the operator spans an analytic semigroup but is not necessarily self-adjoint.
MSC:
| 93B05 | Controllability |
| 35K51 | Initial-boundary value problems for second-order parabolic systems |
| 93C20 | Control/observation systems governed by partial differential equations |
Citations:
Zbl 1203.49011References:
| [1] | W. Alt and C. Schneider, Linear-quadratic control problems with \(L^1\)-control cost, Optimal Control Appl. Methods, 36 (2015), pp. 512-534. · Zbl 1329.49060 |
| [2] | F. Ammar-Khodja, A. Benabdallah, M. González-Burgos, and L. de Teresa, Recent results on the controllability of linear coupled parabolic problems: A survey, Math. Control Relat. Fields, 1 (2011), pp. 267-306. · Zbl 1235.93041 |
| [3] | F. Ammar Khodja, A. Benabdallah, M. González-Burgos, and L. de Teresa, New phenomena for the null controllability of parabolic systems: Minimal time and geometrical dependence, J. Math. Anal. Appl., 444 (2016), pp. 1071-1113. · Zbl 1342.93022 |
| [4] | K. Beauchard, J. Dardé, and S. Ervedoza, Minimal time issues for the observability of Grushin-type equations, Ann. Inst. Fourier (Grenoble), 70 (2020), pp. 247-312. · Zbl 1448.35316 |
| [5] | A. Benabdallah, F. Boyer, and M. Morancey, A block moment method to handle spectral condensation phenomenon in parabolic control problems, Ann. H. Lebesgue, 3 (2020), pp. 717-793. · Zbl 1453.93016 |
| [6] | F. Boyer and P. Fabrie, Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models, Appl. Math. Sci. 183, Springer, New York, 2013. · Zbl 1286.76005 |
| [7] | S. Chowdhury and S. Ervedoza, Open loop stabilization of incompressible Navier-Stokes equations in a \(2\) D channel using power series expansion, J. Math. Pures Appl., 130 (2019), pp. 301-346. · Zbl 1428.35280 |
| [8] | S. Chowdhury, D. Mitra, and M. Renardy, Null controllability of the incompressible Stokes equations in a \(2\)-D channel using normal boundary control, Evol. Equ. Control Theory, 7 (2018), pp. 447-463. · Zbl 1405.93122 |
| [9] | J.-M. Coron and S. Guerrero, Null controllability of the \(N\)-dimensional Stokes system with \(N-1\) scalar controls, J. Differential Equations, 246 (2009), pp. 2908-2921. · Zbl 1172.35042 |
| [10] | J.-M. Coron and P. Lissy, Local null controllability of the three-dimensional Navier-Stokes system with a distributed control having two vanishing components, Invent. Math., 198 (2014), pp. 833-880. · Zbl 1308.35163 |
| [11] | E. B. Davies, Wild spectral behaviour of anharmonic oscillators, Bull. London Math. Soc., 32 (2000), pp. 432-438. · Zbl 1043.47502 |
| [12] | E. B. Davies and A. B. J. Kuijlaars, Spectral asymptotics of the non-self-adjoint harmonic oscillator, J. London Math. Soc. (2), 70 (2004), pp. 420-426. · Zbl 1073.34093 |
| [13] | S. Dolecki, Observability for the one-dimensional heat equation, Studia Math., 48 (1973), pp. 291-305. · Zbl 0264.35036 |
| [14] | M. Duprez and P. Lissy, Positive and negative results on the internal controllability of parabolic equations coupled by zero- and first-order terms, J. Evol. Equ., 18 (2018), pp. 659-680. · Zbl 1396.93021 |
| [15] | E. Fernández-Cara and S. Guerrero, Global Carleman estimates for solutions of parabolic systems defined by transposition and some applications to controllability, AMRX Appl. Math. Res. Express, 2006 (2006), 75090. · Zbl 1113.35037 |
| [16] | E. Fernández-Cara, S. Guerrero, O. Y. Imanuvilov, and J.-P. Puel, Local exact controllability of the Navier-Stokes system, J. Math. Pures Appl. (9), 83 (2004), pp. 1501-1542. · Zbl 1267.93020 |
| [17] | A. V. Fursikov, Exact boundary zero controllability of three-dimensional Navier-Stokes equations, J. Dynam. Control Systems, 1 (1995), pp. 325-350. · Zbl 0951.93005 |
| [18] | A. V. Fursikov and O. Y. Imanuvilov, Controllability of Evolution Equations, Lecture Notes Ser. 34, Seoul National University Research Institute of Mathematics, Global Analysis Research Center, Seoul, 1996. · Zbl 0862.49004 |
| [19] | I. C. Gohberg and M. G. Kreĭn, Introduction to the Theory of Linear Nonselfadjoint Operators, translated from the Russian by A. Feinstein, Transl. Math. Monogr. 18, American Mathematical Society, Providence, RI, 1969. · Zbl 0181.13503 |
| [20] | O. Y. Imanuvilov, Remarks on exact controllability for the Navier-Stokes equations, ESAIM Control Optim. Calc. Var., 6 (2001), pp. 39-72. · Zbl 0961.35104 |
| [21] | K. Ito and K. Kunisch, A variational approach to sparsity optimization based on Lagrange multiplier theory, Inverse Problems, 30 (2014), 015001. · Zbl 1292.65070 |
| [22] | D. Kalise, K. Kunisch, and Z. Rao, Infinite horizon sparse optimal control, J. Optim. Theory Appl., 172 (2017), pp. 481-517. · Zbl 1372.49002 |
| [23] | D. Kalise, K. Kunisch, and Z. Rao, Sparse and switching infinite horizon optimal control with nonconvex penalizations, ESAIM Control Optim. Calc. Var., 26 (2020), 61. · Zbl 1454.49004 |
| [24] | K. Kunisch, P. Trautmann, and B. Vexler, Optimal control of the undamped linear wave equation with measure valued controls, SIAM J. Control Optim., 54 (2016), pp. 1212-1244, https://doi.org/10.1137/141001366. · Zbl 1343.35239 |
| [25] | J.-L. Lions, Remarks on approximate controllability: Festschrift on the occasion of the 70th birthday of Shmuel Agmon, J. Anal. Math., 59 (1992), pp. 103-116. · Zbl 0806.35101 |
| [26] | J.-L. Lions and E. Zuazua, A generic uniqueness result for the Stokes system and its control theoretical consequences, in Partial Differential Equations and Applications, Lecture Notes in Pure and Appl. Math. 177, Dekker, New York, 1996, pp. 221-235. · Zbl 0852.35112 |
| [27] | Y. Liu, T. Takahashi, and M. Tucsnak, Single input controllability of a simplified fluid-structure interaction model, ESAIM Control Optim. Calc. Var., 19 (2012), pp. 20-42. · Zbl 1270.35259 |
| [28] | A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Appl. Math. Sci. 44, Springer-Verlag, New York, 1983. · Zbl 0516.47023 |
| [29] | J.-P. Raymond, Stokes and Navier-Stokes equations with nonhomogeneous boundary conditions, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), pp. 921-951. · Zbl 1136.35070 |
| [30] | M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Birkhäuser Adv. Texts Basler Lehrbücher [Birkhäuser Adv. Texts Basel Textbooks], Birkhäuser Verlag, Basel, 2009. · Zbl 1188.93002 |
| [31] | E. Zuazua, Switching control, J. Eur. Math. Soc. (JEMS), 13 (2011), pp. 85-117. · Zbl 1203.49011 |
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