?-Optimal control of the solution of the evolution equation in a Banach space

1984 ◽  
Vol 36 (2) ◽  
pp. 167-170
Author(s):  
S. A. Mel'nik
Keyword(s):  
Banach Space ◽  
Optimal Control ◽  
Evolution Equation

scholarly journals Admissible relaxation in optimal control problems for infinite dimensional uncertain systems

1992 ◽  
Vol 5 (3) ◽  
pp. 227-236 ◽  
Author(s):  
N. U. Ahmed ◽  
X. Xiang
Keyword(s):  
Banach Space ◽  
Optimal Control ◽  
Evolution Equation ◽  
Uncertain Systems ◽  
Optimal Control Problems ◽  
Original System ◽  
Control Problems ◽  
Infinite Dimensional ◽  
Relaxed System

In this paper we present a result on admissible relaxation for a class of systems governed by an uncertain evolution equation on Banach space. We show that the set of original trajectories is dense in the set of relaxed trajectories and that under certain assumptions the relaxed system is equivalent to the original system.

scholarly journals Asymptotic properties of mild solutions of nonautonomous evolution equations with applications to retarded differential equations

1999 ◽  
Vol 4 (3) ◽  
pp. 169-194 ◽  
Author(s):  
Gabriele Gühring ◽  
Frank Räbiger
Keyword(s):  
Banach Space ◽  
Differential Equations ◽  
Evolution Equation ◽  
Unique Solution ◽  
Evolution Equations ◽  
Bounded Solution ◽  
Mild Solutions ◽  
Asymptotic Properties ◽  
Yosida Operator ◽  
Retarded Differential Equations

We investigate the asymptotic properties of the inhomogeneous nonautonomous evolution equation(d/dt)u(t)=Au(t)+B(t)u(t)+f(t),t∈ℝ, where(A,D(A))is a Hille-Yosida operator on a Banach spaceX,B(t),t∈ℝ, is a family of operators inℒ(D(A)¯,X)satisfying certain boundedness and measurability conditions andf∈L loc 1(ℝ,X). The solutions of the corresponding homogeneous equations are represented by an evolution family(UB(t,s))t≥s. For various function spacesℱwe show conditions on(UB(t,s))t≥sandfwhich ensure the existence of a unique solution contained inℱ. In particular, if(UB(t,s))t≥sisp-periodic there exists a unique bounded solutionusubject to certain spectral assumptions onUB(p,0),fandu. We apply the results to nonautonomous semilinear retarded differential equations. For certainp-periodic retarded differential equations we derive a characteristic equation which is used to determine the spectrum of(UB(t,s))t≥s.

scholarly journals Optimal control of a rate-independent evolution equation via viscous regularization

2017 ◽  
Vol 10 (6) ◽  
pp. 1467-1485 ◽  
Author(s):  
Ulisse Stefanelli ◽  
◽  
Daniel Wachsmuth ◽  
Gerd Wachsmuth ◽  
◽  
...  
Keyword(s):  
Optimal Control ◽  
Evolution Equation ◽  
Independent Evolution ◽  
Viscous Regularization

scholarly journals On the Gevrey ultradifferentiability of weak solutions of an abstract evolution equation with a scalar type spectral operator on the open semi-axis

2019 ◽  
Vol 17 (1) ◽  
pp. 1082-1112
Author(s):  
Marat V. Markin
Keyword(s):  
Banach Space ◽  
Evolution Equation ◽  
Weak Solutions ◽  
A Priori ◽  
Complex Banach Space ◽  
Spectral Operator ◽  
Scalar Type ◽  
Improvement Effect ◽  
Abstract Evolution Equation ◽  
Necessary And Sufficient

Abstract Given the abstract evolution equation $$\begin{array}{} \displaystyle y'(t)=Ay(t),\, t\ge 0, \end{array}$$ with scalar type spectral operator A in a complex Banach space, found are conditions necessary and sufficient for all weak solutions of the equation, which a priori need not be strongly differentiable, to be strongly Gevrey ultradifferentiable of order β ≥ 1, in particular analytic or entire, on the open semi-axis (0, ∞). Also, revealed is a certain interesting inherent smoothness improvement effect.

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