2016/06/12 by Anil Kumar, Kumar, Anil, Amiya K. Pani +3
Computer Science · Engineering · Mathematics · #49J20 #49M30 #65K99 #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #Nonlinear Partial Differential Equations #Optimization and Control (math.OC) #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.1606.03673
openalex publication_date 2016/06/12 · openalex created_date 2022/08/30 · openalex updated_date 2026/07/28
In this paper, we discuss the distributed control problem governed by the\nfollowing parabolic integro-differential equation (PIDE) in the abstract form\n\
frac
partial y
partial t + A y amp;=amp;
int0t B(t, s) y(s)\nds + Gu,
;
, t
in [0, T],
;
;
;
;
;
;
;
;
;
;
;
;
;
;
,
hfill(
ast)
\ny(0) amp;=amp; y0
,
in X,
nonumber where, y denotes the state\nspace variable, u is the control variable, A is a self adjoint, positive\ndefinite linear (not necessarily bounded) operator in a Hilbert space X with\ndense domain D(A) \⊂ X, B(t,s) is an unbounded operator, smooth with\nrespect to t and s with D(A) \⊂ D(B(t,s)) \⊂ X for 0 \≤ s\n\≤ t \≤ T and G is a bounded linear operator from the control space to\nX. Assuming that the corresponding evolution equation (B \≡ 0 in\n(\∗)) is approximately controllable, it is shown that the set of\napproximate controls of the distributed control problem (\∗) is nonempty.\nThe problem is first viewed as constrained optimal control problem and then it\nis approximated by unconstrained problem with a suitable penalty function. The\noptimal pair of the constrained problem is obtained as the limit of optimal\npair sequence of the unconstrained problem. The approximation theorems, which\nguarantee the convergence of the numerical scheme to the optimal pair sequence,\nare also proved.\n