Topological methods for the global controllability of nonlinear systems

Sufficient conditions for the local and global controllability of general nonlinear systems, by means of controls belonging to a fixed finite-dimensional subspace of the space of all admissible controls, are established with the aid of topological methods, such as homotopy invariance principles. Some applications to certain classes of nonlinear control processes are given, and various known results on the controllability of perturbed linear systems are also derived as particular cases.     

Constrained controllability of nonlinear systems

In the paper, infinite-dimensional dynamical systems describedby nonlinear abstract differential equations are considered.Using a generalized open-mapping theorem sufficient conditionsfor constrained exact local controllability are formulated andproved. It is generally assumed that the values of controlsare in a convex and closed cone with the vertex at zero. Asan illustrative example. the constrained exact local controllabilityproblem for nonlinear delayed dynamical system is solved indetail. Some remarks and comments on the existing results forcontrollability of nonlinear dynamical systems are also presented.     

Local controllability for autonomous nonlinear systems

This paper is concerned with the study of local controllability for nonlinear, analytic, autonomous, control systems. To this aim, we consider the control system as a set-valued differential system, and we show that the convexification of the set of velocities does not change the local controllability properties. Given a finite family of vector fields, we also give a necessary condition for local controllability at timet, for everyt.This work was performed under the auspices of the National Research Council of Italy (CNR).     

Relative controllability of perturbations of nonlinear systems

Sufficient conditions are derived for the relative controllability of perturbations of nonlinear systems with distributed delays in the control variables. The results are a generalization of previous results and are obtained by using Schauder's fixed-point theorem.     

Asymptotic null controllability of nonlinear perturbed systems

In this paper, sufficient conditions are obtained for the asymptotic null controllability of the system $$\dot x(t) = g(t,x(t)) + B(t,x(t))u(t) + f(t,x(t),u(t)).$$ The results are obtained by using the Leray-Schauder fixed-point theorem.     

Sample controllability of nonlinear stochastic integrodifferential systems

Sufficient conditions are established for the sample controllability of nonlinear stochastic integrodifferential systems by using the notion of comparison principle and a random fixed point theorem. Examples are provided to illustrate the result.     

The controllability problem for nonlinear Volterra systems

We discuss the controllability of systems whose dynamics are governed by a large class of nonlinear Volterra integral equations. The property of controllability is shown to be equivalent to the existence of a fixed point of a certain set-valued map. We show that convexity and seminormality conditions intimately related to those assumed in proofs of existence theorems for optimal controls are sufficient to guarantee controllability. Approximate controllability results are obtained by first introducing generalized solutions and then showing, under only mild additional restrictions, that ordinary solutions are dense in this broader class of trajectories.Dedicated to L. CesariTu se' lo mio maestro e il mio autore: Tu se' solo colui, da cui io tolsi Lo bello stile che m' ha fatto onore. Dante, Canto I: 85–87This work was written while the author was on leave to the Institut für Numerische und Angewandte Mathematik, Universität Göttingen, Göttingen, West Germany.     

Boundary controllability of Sobolev-type abstract nonlinear integrodifferential systems

Sufficient conditions are established for boundary controllability of various classes of Sobolev-type nonlinear systems including integrodifferential systems in Banach spaces. The results are obtained using the strongly continuous semigroup of operators and the Banach contraction principle. Examples are provided to illustrate the theory.     

Complete controllability of nonlinear stochastic impulsive functional systems

This paper is concerned with the controllability of a kind of nonlinear stochastic impulsive functional systems. Sufficient conditions for the complete controllability of the nonlinear stochastic impulsive functional systems are obtained by using Schauder fixed-point theorem. A numerical example is given to illustrate the effectiveness of the theoretical results.     

On the controllability of nonlinear systems with applications to polynomial systems

The purpose of this paper is to examine the controllability for a class of nonlinear control systems of the form $$\frac{{d\phi t}}{{dt}} = X_0 (\phi t) + \sum\limits_{j = 1}^r {u_j (t)X_j } (\phi _t )$$ Our general theorem is stated in terms of differential geometry, but it will be applied to concrete systems such that coefficientsX ₀,..,X _r are polynomials.     

Sufficient conditions for the controllability of nonlinear distributed systems

For nonlinear distributed systems representable as a Volterra functional operator equation in a Lebesgue space, sufficient conditions for pointwise controllability with respect to a vector of non-linear functionals are proved. The controls are assumed to be piecewise constant vector functions. The reduction of controlled distributed systems to the functional operator equation under study is illustrated by two examples: a Dirichlet boundary value problem for the diffusion equation and a mixed problem for the transport equation.     

Global controllability and computation of controls in nonlinear systems

The authors consider the problem of constructing an admissible open-loop control of bounded energy steering a nonlinear system from a given initial state to a given final state under the condition that the first-approximation system is completely controllable. The convergent iterative procedure for computing an admissible control is verified. It is shown that a nonlinear system locally controllable with respect to the first approximation becomes globally completely controllable for any boundary conditions from the stability region if the initial nonlinear system is stabilizable up to the asymptotic stability in the large and in the whole and the nonlinear terms either satisfy the global Cauchy-Lipschitz condition or are polynomials of a certain degree in state coordinates with arbitrary coefficients. The nonlinear system of algebraic equations to computation of whose solutions the problem of constructing the admissible control reduces is indicated. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 26, Nonlinear Dynamics, 2005.     

On the controllability of some singularly perturbed nonlinear systems

The controllability of a large scale dynamic system which depends singularly upon a small parameter λ is considered. When λ = 0, the large scale system degenerates into a reduced order subsystem representing its slow dynamics while neglecting the fast phenomena. Another subsystem, often called a boundary layer system, represents the fast dynamics. In this paper sufficient conditions are established under which the controllability of the overall large scale system is inferred from the same property of the two subsystems.     

The controllability problem for certain nonlinear intergro-differential volterra systems

The controllability property is proved for so-called "triangular" control systems of integro-differential Volterra equations. This result generalizes the previous results obtained for the case of control systems of ordinary differential equations. Two special cases, when the tools developed for the differential systems work without using the general result of this paper are considered.     

Functional inclusion and controllability of nonlinear neutral functional differential systems

Consider the nonlinear neutral functional differential inclusion (i) $$(d/dt)D(t, x_t ) \in R(t, x_t )$$ , whereD is a continuous operator onIXC, linear inx _t , indeed of the form (4) below, with kernelD(t, ·)={0}, and atomic at 0, andR is nonempty, closed, and convex. Here,I≡[t ₀,t _I ] andC=C([-h,0],E ⁿ ). In (i), the derivative is specified in terms of the state at timet as well as the state and the derivative of the state for values oft precedingt. We use the Fan fixed-point theorem to prove the existence of a solution of (i) which satisfies two-point boundary values \(x_\omega = \phi _0 ,x_{t_1 } = \phi _1\) , where φ₀, φ₁ belong toC. We next apply this existence result to study the exact function space controllability of the neutral functional differential system (ii) $$(d/dt)D(t, x_t ) = f(t, x_t , u), u(t) \in \Omega (t, x_t )$$ . We present sufficient conditions onf and Ω which imply exact controllability between two fixed functions inC.