Exact controllability of infinite dimensional systems with controls of minimal norm

Luisa Malaguti, Stefania Perrotta, Valentina Taddei

DOI: http://dx.doi.org/10.12775/TMNA.2019.087


The paper deals with the exact controllability of a semilinear system in a separable Hilbert space. A bounded linear part is considered and a linear control introduced. The state space is compactly embedded in a Banach space and the nonlinear term is continuous in its state variable in the norm of the Banach space. An infinite sequence of finite dimensional controllability problems is introduced and the solution is obtained by a limiting procedure. To the best of our knowledge, the method is new in controllability theory. An application to an integro-differential system in euclidean spaces completes the discussion.


Controllability; fixed point; approximation solvability method

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K. Balachandran and J.P. Dauer, Controllability of nonlinear systems in Banach spaces: a survey, Dedicated to Professor Wolfram Stadler, J. Optim. Theory Appl. 115 (2002), 7–28.

V. Barbu, Nonlinear semigroups and differential equations in Banach spaces, Nordoof International Publishing, Leyden, 1976.

P.W. Bates and G. Zhao, Existence, uniqueness and stability of the stationary solution to a nonlocal evolution equation arising in population dispersal, J. Math. Anal. Appl. 332 (2007), 428–440.

I. Benedetti, N.V. Loi, L. Malaguti and V. Obukhovskiı̆, An approximation solvability method for nonlocal differential problems in Hilbert spaces, Commun. Contemp. Math. 19 (2017), 1–33.

I. Benedetti, N.V. Loi, L. Malaguti and V. Taddei, Nonlocal diffusion second order partial differential equations, J. Differential Equations 262 (2017), 1499–1523.

I. Benedetti, V. Obukhovskiı̆ and V. Taddei, Controllability for systems governed by semilinear evolution inclusions without compactness, NoDEA Nonlinear Differential Equations Appl. 21 (2014), 795–812.

M.D. Burlică, M. Necula, D. Roşu and I.I. Vrabie, Delay Differential Evolutions Subjected to Nonlocal Initial Conditions, Monographs and Research Notes in Mathematics, CRC Press, Boca Raton, FL, 2016.

R. Curtain and H. Zwart, An Introduction to Infinite-Dimensional Linear Systems Theory, Texts in Applied Mathematics, vol. 21, Springer, New York, 1995.

J. Diestel and J.J. Uhl Jr, Vector Measures, Math. Surveys, vol. 15, AMS, Providence, RI, 1977.

M. Kamenskiı̆, V. Obukhovskiı̆ and P. Zecca, Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Space, Grundlehren Math. Wiss., W. de Gruyter, Berlin, 2001.

K. Magnusson, J. Pritchard and M.D. Quinn, The application of fixed point theorems to global nonlinear controllability problems, Banach Center Publ. 14 (1985), 319–344.

V. Maz’ya, Sobolev Spaces, Grundlehren Math. Wiss., vol. 342, Springer, Berlin, 2011.

V. Obukhovskiı̆ and P. Zecca, Controllability for systems governed by semilinear differential inclusions in a Banach space with a noncompact semigroup, Nonlinear Anal. 70 (2009), 3424–3436.

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, vol. 44, Springer, New York, 1983.

A.E. Taylor and D.C. Lay, Introduction to Functional Analysis, John Wiley & Sons Inc, New York, 1980.

R. Triggiani, A note on the lack of exact controllability for mild solutions in Banach spaces, SIAM J. Control Optim. 15 (1977), 407–411.

M. Tucsnak and G. Weiss, Observation and control for operator semigroups, Birkhäuser Advanced Texts, Birkhäuser, Basel, 2009.

I.I. Vrabie, C0 Semigroups and Applications, North-Holland Mathematics Studies, vol. 191, North-Holland Publishing Co., Amsterdam, 2003.

R. Zawiski, Exact controllability of non-Lipschitz semilinear systems, J. Fixed Point Theory Appl. 20 (2018), pp. 67, DOI: 10.1007/s11784-018-0550-5.


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