Controllability of coupled parabolic systems with multiple underactuations. II: Null controllability. (English) Zbl 1422.35108
Summary: This paper is the second of two parts which together study the null controllability of a system of coupled parabolic PDEs. Our work specializes to an important subclass of these control problems which are coupled by first- and zero-order couplings and are, additionally, underactuated. In the first part of our work [ibid. 57, No. 5, 3272–3296 (2019; Zbl 1422.35107)], we posed our control problem in a framework which divided the problem into interconnected components: the algebraic control problem, which was the focus of the first part, and the analytic control problem, whose treatment was deferred to this paper. We use slightly nonclassical techniques to prove null controllability of the analytic control problem by means of internal controls appearing on every equation. We combine our previous results in [loc. cit.] with the ones derived below to establish a null controllability result for the original problem.
Citations:
Zbl 1422.35107References:
| [1] | J. Conway, A Course in Functional Analysis, 2nd ed., Grad. Texts Math. 96, Springer, New York, 1990. · Zbl 0706.46003 |
| [2] | J.-M. Coron, Control and Nonlinearity, Math. Surveys Monogr. 136, American Mathematical Society, Providence, RI, 2007. · Zbl 1140.93002 |
| [3] | 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, https://doi.org/10.1016/j.jde.2008.10.019. · Zbl 1172.35042 |
| [4] | 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, https://doi.org/10.1007/s00222-014-0512-5. · Zbl 1308.35163 |
| [5] | M. Duprez and P. Lissy, Indirect controllability of some linear parabolic systems of \(m\) equations with m-1 controls involving coupling terms of zero or first order, J. Math. Pures Appl. (9), 106 (2016), pp. 905-934, https://doi.org/10.1016/j.matpur.2016.03.016. · Zbl 1350.93016 |
| [6] | L. Evans, Partial Differential Equations, 2nd ed., Grad. Stud. Math. 19, American Mathematical Society, Providence, RI, 2010, https://doi.org/10.1090/gsm/019. · Zbl 1194.35001 |
| [7] | E. Fernández-Cara, M. González-Burgos, S. Guerrero, and J.-P. Puel, Null controllability of the heat equation with boundary Fourier conditions: The linear case, ESAIM Control Optim. Calc. Var., 12 (2006), pp. 442-465, https://doi.org/10.1051/cocv:2006010. · Zbl 1106.93009 |
| [8] | A. Fursikov and O. Imanuvilov, Controllability of Evolution Equations, Lecture Notes Ser. 34, Seoul National University, Seoul, 1996. · Zbl 0862.49004 |
| [9] | J.-L. Lions, Contrôle Optimal de Systemes Gouvernés par des Équations aux Dérivées Partielles, Dunod, Paris, 1968. · Zbl 0179.41801 |
| [10] | L. Nirenberg, On elliptic partial differential equations, Ann. Sc. Norm. Super. Pisa (3), 13 (1959), pp. 115-162. · Zbl 0088.07601 |
| [11] | D. Steeves, B. Gharesifard, and A.-R. Mansouri, Controllability of coupled parabolic systems with multiple underactuations, part 1: Algebraic solvability, SIAM J. Control Optim., 57 (2019), pp. 3272-3296. · Zbl 1422.35107 |
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