Abstract
This paper presents a variational method for studying approximate controllability and infinite-dimensional exact controllability (finite-approximate controllability) for Riemann–Liouville fractional linear/semilinear evolution equations in Hilbert spaces. A useful criterion for finite-approximate controllability of Riemann–Liouville fractional linear evolution equations is formulated in terms of resolvent-like operators. We also find that such a control provides finite-dimensional exact controllability in addition to the approximate controllability requirement. Assuming the finite-approximate controllability of the corresponding linearized RL fractional evolution equation, we obtain sufficient conditions for finite-approximate controllability of the semilinear RL fractional evolution equation under natural conditions. The results are a generalization and continuation of recent results on this subject. Applications to fractional heat equations are considered.
Highlights
Controllability is one of the fundamental qualitative concepts in modern mathematical control theory, which plays an important role in deterministic/stochastic control theory of dynamical systems
Exact controllability allows the system to be controlled to an arbitrary final state, while approximate controllability means that the system can be controlled to an arbitrary small neighborhood of the finite state, and very often approximate controllability is quite sufficient in applications
We investigate simultaneous approximate and finite-dimensional exact controllability of the following RL fractional semilinear evolution system: RL D α y ( t )
Summary
Controllability is one of the fundamental qualitative concepts in modern mathematical control theory, which plays an important role in deterministic/stochastic control theory of dynamical systems. Dauer and Mahmudov [22] and Mahmudov [10] have used a resolvent approach, used by Bashirov and Mahmudov [3] to study approximate controllability for linear evolution equations, and obtained some sufficient conditions for the approximate controllability of classical semilinear systems. The main contributions of this work can be summarized as follows: We develop a constructive variational approach that is somewhat different from approaches used in the literature, and provide a necessary and sufficient condition for the finite-approximate controllability of linear classical/fractional evolution systems in terms of resolvent-like operators (Criterion (iv) of Theorem 1). We define resolvent-like operators and give necessary and sufficient conditions for finite-approximate controllability of classical/fractional linear evolution equations. We present two examples to demonstrate our main results
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