Existence and Optimal Control Results for Second-Order Semilinear System in Hilbert Spaces

• Published in: Circuits, systems, and signal processing Vol. 40; no. 9; pp. 4246 - 4258
• Main Authors: Shukla, Anurag, Patel, Rohit
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.09.2021; Springer Nature B.V
• Subjects: Banach spaces; Circuits and Systems; Control systems; Control theory; Cost function; Dynamical systems; Electrical Engineering; Electronics and Microelectronics; Engineering; Fixed points (mathematics); Hilbert space; Instrumentation; Optimal control; Signal processing; Signal,Image and Speech Processing; Sine and cosine family; Optimal control; Mild solution; 34H05; 34K35; Second-order system; 49J15
• ISSN: 0278-081X; 1531-5878
• DOI: 10.1007/s00034-021-01680-2

The article provides sufficient conditions for the existence of optimal control for second-order semilinear control system in Hilbert spaces. We consider the integral cost function as J ( z , v ) : = ∫ 0 T L ( τ , z v ( τ ) , v ( τ ) ) d t , subject to the equations z ′ ′ ( τ ) = A z ( τ ) + B v ( τ ) + g ( τ , z ( τ ) ) ; 0 < τ ≤ T z ( 0 ) = z 0 z ′ ( 0 ) = z 1 . Next, we discuss the existence and the uniqueness of mild solutions for the above proposed problem using Banach fixed point theorem. The stated Lagrange’s problem admits at least one optimal control pair under certain assumptions. Finally, the validation of theoretical results is provided through an example.