Abstract
We consider problems of control and problems of optimal control, monitored by an abstract equation of the formEx=N u x in a finite interval [0,T]; here,x is the state variable with values in a reflexive Banach space;u is the control variable with values in a metric space;E is linear and monotone; andN u is nonlinear of the Nemitsky type. Thus, by well-known devices, the results apply also to parabolic partial differential equations in a cylinder [0,T]×G,G ⊂ ℝn, with Cauchy data fort=0 and Dirichlet or Neumann conditions on the lateral surface of the cylinder. We prove existence theorems for solutions and existence theorems for optimal solutions, by reduction to a theorem of Kemochi for reflexive Banach spaces.
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Hou, S.H. Existence of solutions for a class of nonlinear control problems. J Optim Theory Appl 66, 23–35 (1990). https://doi.org/10.1007/BF00940530
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DOI: https://doi.org/10.1007/BF00940530