Sufficient Conditions For Local Controllability Research Articles

A sufficient condition for local controllability of a Caputo type fractional differential inclusion

Abstract We consider a Cauchy problem for a fractional differential inclusion defined by a Caputo type fractional derivative and we obtain a sufficient condition for local controllability along a reference trajectory in terms of a certain fractional variational differential inclusion associated to the initial problem.

Local Controllability of Single-Input Nonlinear Systems Based on Deterministic Wiener Processes

This technical note proposes a new sufficient condition for local controllability of nonlinear control systems ensured by single inputs. The objective is achieved by allowing control inputs to have the limitation of the derivation of unbounded-variation functions as with white noises, provided that we just consider deterministic signals. The treatment of the signals is mathematically recognized via rough path analysis, and then the dynamics of the target systems are represented by rough differential equations as the extension of ordinary differential equations. Our sufficient condition is clarified by deterministic Wiener processes that produce an effect on the dynamics of the systems similar to Wiener processes despite the nonprobabilistic property. The novelty of our condition is confirmed by a single-input two-link underactuated robot arm moving on the horizontal plane.

Local Controllability for Abnormal Systems

This paper puts forward sufficient conditions for local controllability of abnormal systems, that is, in the case when a linearization of a system at the admissible process under consideration is not controllable. Sufficient conditions for local controllability of a nonlinear system at an equilibrium point are obtained as an application of these results. The proof of the sufficient conditions for local controllability is based on the implicit function theorem, which is meaningful near a singular point of the mapping defining the equation. This theorem is proved in the second section.

Metrically Regular Differential Generalized Equations

In this paper we consider a control system coupled with a generalized equation, which we call a differential generalized equation (DGE). This model covers a large territory in control and optimization, such as differential variational inequalities, control systems with constraints, as well as necessary optimality conditions in optimal control. We study metric regularity and strong metric regularity of mappings associated with DGE by focusing in particular on the interplay between the pointwise versions of these properties and their infinite-dimensional counterparts. Metric regularity of a control system subject to inequality state-control constraints is characterized. A sufficient condition for local controllability of a nonlinear system is obtained via metric regularity. Sufficient conditions for strong metric regularity in function spaces are presented in terms of uniform pointwise strong metric regularity. A characterization of the Lipschitz continuity of the control part of the solution mapping as a function of time is established. Finally, a path-following procedure for a discretized DGE is proposed for which an error estimate is derived.

Relaxation and controllability in optimal control problems

The paper is concerned with the relationships between necessary minimum conditions in an optimal control problem, minimum conditions in the corresponding relaxed problem and sufficient conditions for local controllability of the system. The results obtained are applied to a fairly general optimal control problem. Bibliography: 7 titles.

Local Controllability of Models of Combined Anticancer Therapy with Delays in Control

We present sufficient conditions of local controllability for a class of models of treatment response to combined anticancer therapies which include delays in control strategies. The combined therapy is understood as combination of direct anticancer strategy e.g. chemotherapy and indirect modality (in this case antiangiogenic therapy). Controllability of the models in the form of semilinear second order dynamic systems with delays in control enables to answer the questions of realizability of different objectives of multimodal therapy in the presence of PK/PD effects. We compare results for the models without delays and conditions for relative local controllability of models with delays.

Comparison of Controllability Conditions for Models of Antiangiogenic and Combined Anticancer Therapy

We compare sufficient conditions for local controllability for a class of models of treatment response to antiangiogenic and combined anticancer therapies. The combined therapy is understood as combination of direct anticancer strategy e.g. chemotherapy and indirect modality (in this case antiangiogenic therapy). We discuss different control objectives related to multimodal therapies in the light of present knowledge of self-organization of tumor cell populations treated by anticancer drugs. Controllability of the models in the form of semilinear second order dynamic systems enables to expect that the objectives could be reached.

Local Controllability around Closed Orbits

We give a necessary and sufficient condition for local controllability around closed orbits for general smooth control systems. We also prove that any such system on a compact manifold has a closed orbit.

Controllability of a class of switching control systems and application to the Kinetic Battery Model

This paper is concerned with the controllability of a class of switching control systems in which the zero control is an extreme point of the control constraint set. Sufficient and necessary conditions for local controllability at the origin of such systems are presented. In addition, sufficient conditions for local controllability at a nonzero state of such systems are established. Finally, we use the obtained results to study the controllability of the Kinetic Battery Model.

Equilibrium control of nonlinear verticum-type systems, applied to integrated pest control

Linear verticum-type control and observation systems have been introduced for modelling certain industrial systems, consisting of subsystems, vertically connected by certain state variables. Recently the concept of verticum-type observation systems and the corresponding observability condition have been extended by the authors to the nonlinear case. In the present paper the general concept of a nonlinear verticum-type control system is introduced, and a sufficient condition for local controllability to equilibrium is obtained. In addition to a usual linearization, the basic idea is a decomposition of the control of the whole system into the control of the subsystems. Starting from the integrated pest control model of Rafikov and Limeira (2012) and Rafikov et al. (2012), a nonlinear verticum-type model has been set up an equilibrium control is obtained. Furthermore, a corresponding bioeconomical problem is solved minimizing the total cost of integrated pest control (combining chemical control with a biological one).

Set separation, approximating multicones, and the Lipschitz maximum principle

We present a general necessary condition for separation of the reachable set of a Lipschitz control system from another given set of states, expressed in terms of an “approximating multicone” to the set in a sense that contains as special cases the Clarke and Mordukhovich cones. We then show how this separation result implies a strengthened form of the usual sufficient condition for local controllability along the reference curve and the necessary condition for optimality.

Minimum principle and controllability for multiparameter discrete inclusions via derived cones

We consider a multiparameter discrete inclusion and we prove that the reachable set of a certain variational multiparameter discrete inclusion is a derived cone in the sense of Hestenes to the reachable set of the discrete inclusion. This result allows to obtain sufficient conditions for local controllability along a reference trajectory and a new proof of the minimum principle for an optimization problem given by a multiparameter discrete inclusion with endpoint constraints.

A polynomial approach to discrete-time non-linear system controllability

This paper uses a polynomial approach to present a necessary and sufficient condition for local controllability of single-input single-output (SISO) discrete-time non-linear systems. The condition is presented in terms of common factors of a non-commutative polynomial expression. This extends the result on controllability of linear systems and the recent result on controllability of continuous-time non-linear systems to expose controllability of discrete-time non-linear systems in the input–output framework. The controllability condition yields a finite step computing procedure for examining controllability of discrete-time input–output non-linear systems.

On Local Fixed Point Controllability of Nonlinear Difference Equations

In this note we consider nonlinear systems of difference equations of the form where is a continuous mapping and and are state and control functions, respectively. We derive sufficient conditions for local controllability into fixed points of the uncontrolled system i.e. solutions of the equation The result is applied to a time-discrete predator–prey model.

Optimization of Mean Fitness of a Population via Artificial Selection

The proposed control-theoretical model of artificial selection is based on the classical Fisher model of natural selection. Artificial selection is realized by changing the fitness of certain genotypes in a Mendelian population. Time-dependent fitness parameters are considered as control functions (artificial selection strategies). Under certain conditions on the model parameters the mean fitness of the population attains a maximum at the equilibrium of the selection dynamics. Thus the problem of optimization of the mean fitness via artificial selection is an optimal control problem. Using a sufficient condition for local controllability of nonlinear systems with invariant manifold, the existence of optimal artificial selection strategy is obtained.

Controllability of selection–mutation systems

The well-known Fisher type selection–mutation model is studied from the point of view of mathematical systems theory. Mutation rates are considered as control functions. Based on a general sufficient condition for local controllability of non-linear systems with invariant manifold, a method is proposed to guarantee the controllability of the considered population into a polymorphic equilibrium.

Local Controllability of Quasilinear Integrodifferential Evolution Systems in Banach Spaces

Sufficient conditions for local controllability of quasilinear integrodifferential systems in Banach spaces are established. The results are obtained by using the analytic semigroup theory and the Schauder fixed-point theorem. An example is provided to illustrate the theory.

A polynomial approach to nonlinear system controllability

Uses a polynomial approach to present a necessary and sufficient condition for local controllability of single-input-single-output (SISO) nonlinear systems. The condition is presented in terms of common factors of a noncommutative polynomial expression. This result exposes controllability properties of a nonlinear system in the input-output framework, and gives a computable procedure for examining nonlinear system controllability using computer algebra.

Sufficient conditions for local controllability and high-order necessary conditions for optimality. A differential-geometric approach

Sufficient conditions for local controllability and high-order necessary conditions for optimality. A differential-geometric approach

Controllability Along a Trajectory: A Variational Approach

This paper unifies and improves most sufficient conditions of local controllability both along a trajectory and at a point. This is accomplished by defining high-order variations that can be continuously summed. The peculiar property of these variations is that they may be generated by thin conditions as relations in the Lie algebra associated with a control system. This property leads to a variational interpretation of Sussmann-type sufficient conditions of local controllability (neutralization of obstructions), and it allows the use of different weights for neutralization.