Infinite Dimensional Control

The study of infinite dimensional optimal control has been a rapidly growing area of research (Ahmed, Comm Appl Nonlinear Anal 16:1-14, 2009; Ahmed, Discuss Math Differ Incl Control Optim 35:65-87, 2015; Ahmed, J Abstr Differ Equ Appl 7:11-29, 2016; Ahmed and Xiang, Nonlinear Funct Anal Appl 8:461–488, 2003; Bachir and Blot, Set-Valued Var Anal 23:43–54, 2015; Bachir and Blot, Pure Appl Funct Anal 2:411–426, 2017; Ball, Proc Amer Math Soc 63:370–373, 1977; Barbu, Optimal control of variational inequalities. Pitman Research Notes in Mathematics, London Boston, 1984; Barbu, Analysis and control of nonlinear infinite dimensional systems. Academic Press, Boston New York, 1993; Barbu and Precupanu, Convexity and optimization in Banach spaces. Springer Monographs in Mathematics, Springer Dordrecht Heidelberg London New York, 2012; Coron, Control and nonlinearity. AMS, Providence, Rhode Island, 2007; Gugat et al., Systems Control Lett 90:61–70, 2016; Jiang et al., Appl Anal 96:2349–2366, 2017; Kien et al., SIAM J Control Optim 50:2889–2906, 2012; Lasiecka and Triggiani, Control theory for partial differential equations: continuous and approximation theories; Vol 1: Abstract parabolic systems. Encyclopedia of Mathematics and Its Applications Series, Cambridge University Press, 2000; Lasiecka and Triggiani, Control theory for partial differential equations: continuous and approximation theories; Vol 2: Abstract hyperbolic-like systems over a finite time horizon. Encyclopedia of Mathematics and Its Applications Series, Cambridge University Press, 2000; Li and Yong, Optimal control theory for infinite dimensional systems. Birkhauser, Boston Basel Berlin, 1995; Mordukhovich, Appl Anal 90:1075–1109, 2011; Mordukhovich and Shvartsman, Optimization and feedback control of constrained parabolic systems under uncertain perturbations, Optimal control, stabilization and nonsmooth analysis. Lecture Notes Control Inform. Sci., Springer:121–132, 2004; Porretta and Zuazua, SIAM J Control Optim 51, 4242–4273, 2013; Trelat et al., Pure Appl Funct Anal 3:255–269, 2018; Trelat et al., SIAM J Control Optim 56:1222–1252, 2018; Troltzsch, Optimal control of partial differential equations. Theory, methods and applications. American Mathematical Society, Providence, RI, 2010; Tucsnak and Weiss, Observation and control for operator semigroups. Birkhauser, Basel, 2009; Zaslavski, Appl Math Optim 42:291–313, 2000). In this chapter we present preliminaries which we need in order to study turnpike properties of infinite dimensional optimal control problems. We discuss unbounded operators, C 0 $$C_0$$ semigroups, evolution equations, and admissible control operators.