Controllability Of Semilinear Systems Research Articles

Time Optimal Control of Semilinear Control Systems Involving Time Delays

This paper investigates the time optimal control problem to a target set for semilinear control systems involving time delays or memories when a principal operator is unbounded by the construction of a fundamental solution and an easy consequence of the definition of real interpolation spaces. A convergence theorem of time optimal controls for the given semilinear retarded system to a point target set is also given.

Controllability of second-order semilinear fuzzy integrodifferential control systems with nonlocal initial conditions

This paper is concerned with the nonlocal controllability for the second-order semilinear fuzzy integrodifferential control systems with nonlocal initial conditions. The results are obtained using the Banach fixed point theorem and the fuzzy number whose values are normal, convex, upper semicontinuous and compactly supported interval in $$E_N$$EN. An example is given to illustrate the results.

Universal regular control for generic semilinear systems

We consider discrete-time projective semilinear control systems \(\xi _{t+1} = A(u_t) \cdot \xi _t\), where the states \(\xi _t\) are in projective space \(\mathbb {R}\hbox {P}^{d-1}\), inputs \(u_t\) are in a manifold \(\mathcal {U}\) of arbitrary finite dimension, and \(A :\mathcal {U}\rightarrow \hbox {GL}(d,\mathbb {R})\) is a differentiable mapping. An input sequence \((u_0,\ldots ,u_{N-1})\) is called universally regular if for any initial state \(\xi _0 \in \mathbb {R}\hbox {P}^{d-1}\), the derivative of the time-\(N\) state with respect to the inputs is onto. In this paper, we deal with the universal regularity of constant input sequences \((u_0, \ldots , u_0)\). Our main result states that generically in the space of such systems, for sufficiently large \(N\), all constant inputs of length \(N\) are universally regular, with the exception of a discrete set. More precisely, the conclusion holds for a \(C^2\)-open and \(C^\infty \)-dense set of maps \(A\), and \(N\) only depends on \(d\) and on the dimension of \(\mathcal {U}\). We also show that the inputs on that discrete set are nearly universally regular; indeed, there is a unique non-regular initial state, and its corank is 1. In order to establish the result, we study the spaces of bilinear control systems. We show that the codimension of the set of systems for which the zero input is not universally regular coincides with the dimension of the control space. The proof is based on careful matrix analysis and some elementary algebraic geometry. Then the main result follows by applying standard transversality theorems.

Controllability of fractional order system with nonlinear term having integral contractor

Abstract In this paper, controllability results for a class of semilinear control systems of fractional order are established. The nonlinear term is assumed to have an integral contractor which is a weaker condition than the Lipschitz continuity. The existence and uniqueness of mild solution is also proved.

Approximate Controllability of Systems Determined by Almost Sectorial Operators

In this work we study the approximate controllability of first order linear and semi-linear abstract control systems described by an almost sectorial operator.

Approximate Controllability of Fractional Order Retarded Semilinear Control Systems

In this paper, approximate controllability of fractional order retarded semilinear systems is studied when the nonlinear term satisfies the newly formulated bounded integral contractor-type conditions. We have shown the existence and uniqueness of the mild solution for the fractional order retarded semilinear systems using an iterative procedure approach. Finally, we obtain the approximate controllability results of the system under simple condition.

On Partial Complete Controllability of Semilinear Systems

Many control systems can be written as a first-order differential equation if the state space enlarged. Therefore, general conditions on controllability, stated for the first-order differential equations, are too strong for these systems. For such systems partial controllability concepts, which assume the original state space, are more suitable. In this paper, a sufficient condition for the partial complete controllability of semilinear control system is proved. The result is demonstrated through examples.

Approximate controllability of a delayed semilinear control system with growing nonlinear term

This paper deals with the approximate controllability of a semilinear control system with delay, where the nonlinear term satisfies the linear growth condition. Sufficient conditions for the approximate controllability of a semilinear control system have been established by assuming that the corresponding linear control system with delay is approximately controllable. To prove our results, the Schauder fixed point theorem is applied and instead of a Co-semigroup associated with the mild solution of the system we use the so-called fundamental solution.

Viability of stochastic semi-linear control systems via the quasi-tangency condition

In this paper, we study a criterion for the viability of stochastic semi-linear control systems on a real, separable Hilbert space. The necessary and sufficient conditions are given using the notion of stochastic quasi-tangency. As a consequence, we prove that approximate viability and the viability property coincide for stochastic linear control systems. We obtain Nagumo's stochastic theorem and we present a method allowing to provide explicit criteria for the viability of smooth sets. We analyse the conditions characterizing the viability of the unit ball. The paper generalizes recent results from the deterministic framework.

Existence and stabilizability of steady-state for semilinear pulse-width sampler controlled system

In this paper, we study the steady-state of a semilinear pulse-width sampler controlled system on infinite dimensional spaces. Firstly, by virtue of Schauder's fixed point theorem, the existence of periodic solutions is given. Secondly, utilizing a generalized Gronwall inequality given by us and the Banach fixed point theorem, the existence and stabilizability of a steady-state for the semilinear control system with pulse-width sampler is also obtained. At last, an example is given for demonstration.

S-Controllability of an Abstract First Order Semilinear Control System

The objective of this article is to investigate some sufficient conditions, for S-controllability of an abstract first order semilinear control system of the form Here, A is the closed linear operator, B and C are bounded linear operators defined on suitable Hilbert spaces, ω and ν are suitable wiener processes, and d is the stochastic differentiation. The results are obtained by the separation principle approach. An example is provided to illustrate the technique.

Approximate controllability results of semilinear integrodifferential equations with infinite delays

We study the approximate controllability of control systems governed by a class of semilinear integrodifferential equations with infinite delays. Sufficient conditions for approximate controllability of semilinear control systems are established provided the approximate controllability of the corresponding linear control systems. The results are obtained by using fixed-point theorems and semigroup theory. As an illustration of the application of the approximate controllability results, a simple example is provided.

Constrained Controllability of Second Order Dynamical Systems with Delay

In the paper finite-dimensional dynamical control systems described by second order semilinear stationary ordinary differential state equations with delay in control are considered. Using a generalized open mapping theorem, sufficient conditions for constrained local controllability in a given time interval are formulated and proved. These conditions require verification of constrained global controllability of the associated linear first-order dynamical control system. It is generally assumed, that the values of admissible controls are in a convex and closed cone with vertex at zero. Moreover, several remarks and comments on the existing results for controllability of semilinear dynamical control systems are also presented. Finally, simple numerical example which illustrates theoretical considerations is also given. It should be pointed out, that the results given in the paper extend for the case of semilinear second-order dynamical systems constrained controllability conditions, which were previously known only for linear second-order systems.

Constrained Controllability of Second Order Semilinear Systems

Abstract In the paper finite-dimensional dynamical control systems described by second order semilinear stationary ordinary differential state equations are considered. Using a generalized open mapping theorem, sufficient conditions for constrained local controllability in a given time interval are formulated and proved. These conditions require verification of constrained global controllability of the associated linear first-order dynamical control system. It is generally assumed, that the values of admissible controls are in a convex and closed cone with vertex at zero. Moreover, several remarks and comments on the existing results for controllability of semilinear dynamical control systems are also presented. Finally, simple numerical example, which illustrates theoretical considerations is also given. It should be pointed out, that the results given in the paper extend for the case of semilinear second-order dynamical systems constrained controllability conditions, which were previously known only for linear second-order systems.

On the minimum time function and the minimum energy problem; a nonlinear case

For a semilinear control system in general Banach spaces, we prove results on exact and approximate controllability, regularity of the minimum time function, connections between the minimum time function and the minimum energy.

Approximate controllability of delayed semilinear control systems

We deal with the approximate controllability of control systems governed by delayed semilinear differential equations . Various sufficient conditions for approximate controllability have been obtained; these results usually require some complicated and limited assumptions. Results in this paper provide sufficient conditions for the approximate controllability of a class of delayed semilinear control systems under natural assumptions.

Controllability of Some Nonlinear Systems in Hilbert Spaces

In this paper, several abstract results concerning the controllability of semilinear evolution systems are obtained. First, approximate controllability conditions for semilinear systems are obtained by means of a fixed-point theorem of the Rothe type; in this case, the compactness of the linear operator is assumed. Next, the exact controllability of semilinear systems with nonlinearities having small Lipschitz constants is derived by means of the Banach fixed-point theorem; in this case, the compactness of the operators is not assumed. In both cases, it is proven that the controllability of the linear system implies the controllability of the associated semilinear system. Finally, these abstract results are applied to the controllability of the semilinear wave and heat equations.

On the Bellman Equation for the Minimum Time Problem in Infinite Dimensions

We consider the time optimal control problem for a semilinear parabolic control system, where the target is the closed ball with center 0 and radius $Rgeq0$ in a Hilbert space X. In particular, we allow the origin of X to be the target. Using an appropriate Kruzkov-type transformation, we give an existence and uniqueness result for the associated Hamilton--Jacobi--Bellman equation, even when the reachable set is not the whole space.

Approximate controllability of semilinear neutral systems in Hilbert spaces

The approximate controllability of semilinear neutral systems in Hilbert spaces is studied using the Schauder fixed point theorem. It is shown that the approximate controllability of the semilinear system under some conditions is implied by the approximate controllability of its linear part.

Approximate Controllability of Semilinear Deterministic and Stochastic Evolution Equations in Abstract Spaces

Various sufficient conditions for approximate controllability of linear evolution systems in abstract spaces have been obtained, but approximate controllability of semilinear control systems usually requires some complicated and limited assumptions. In this paper, we show the approximate controllability of the abstract semilinear deterministic and stochastic control systems under the natural assumption that the associated linear control system is approximately controllable. The results are obtained using new properties of symmetric operators (which are proved in this paper), compact semigroups, the Schauder fixed point theorem, and/or the contraction mapping principle.