scholarly journals Decidability of hybrid systems with rectangular differential inclusions

1994 ◽  
pp. 95-104 ◽  
Author(s):  
Anuj Puri ◽  
Pravin Varaiya
Keyword(s):  
Hybrid Systems ◽  
Differential Inclusions

Preliminaries from set-valued analysis

2012 ◽  
Author(s):  
Rafal Goebel ◽  
Ricardo G. Sanfelice ◽  
Andrew R. Teel
Keyword(s):  
Hybrid Systems ◽  
Differential Inclusions ◽  
Set Valued Mapping ◽  
Continuity Properties ◽  
Set Valued Mappings

This chapter includes the necessary background on further developments in the theory of hybrid systems. It first presents the notion of convergence for a sequence of sets and how it generalizes the notion of convergence of a sequence of points. The chapter then deals with set-valued mappings and their continuity properties. Given a set-valued mapping M : ℝᵐ ⇉ ℝⁿ, the chapter defines the range of M as the set rgeM = {‎y ∈ ℝⁿ : Ǝx ∈ ℝᵐ such that y ∈ M(x)}‎; and the graph of M as the set gphM = {‎(x,y) ∈ ℝᵐ × ℝⁿ : y ∈ M(x)}‎. The chapter also specializes some of the concepts, such as graphical convergence, to hybrid arcs and provides further details in such a setting. Finally, the chapter discusses differential inclusions.

Impulse differential inclusions: a viability approach to hybrid systems

2002 ◽  
Vol 47 (1) ◽  
pp. 2-20 ◽  
Author(s):  
J.-P. Aubin ◽  
J. Lygeros ◽  
M. Quincampoix ◽  
S. Sastry ◽  
N. Seube
Keyword(s):  
Hybrid Systems ◽  
Differential Inclusions

scholarly journals Local analysis of hybrid systems on polyhedral sets with state-dependent switching

2014 ◽  
Vol 24 (2) ◽  
pp. 341-355 ◽  
Author(s):  
John Leth ◽  
Rafael Wisniewski
Keyword(s):  
Stability Analysis ◽  
Hybrid Systems ◽  
Hybrid System ◽  
Equivalence Relation ◽  
Stability Theory ◽  
Differential Inclusions ◽  
Local Analysis ◽  
Polyhedral Sets ◽  
State Dependent ◽  
Stability Concepts

Abstract This paper deals with stability analysis of hybrid systems. Various stability concepts related to hybrid systems are introduced. The paper advocates a local analysis. It involves the equivalence relation generated by reset maps of a hybrid system. To establish a tangible method for stability analysis, we introduce the notion of a chart, which locally reduces the complexity of the hybrid system. In a chart, a hybrid system is particularly simple and can be analyzed with the use of methods borrowed from the theory of differential inclusions. Thus, the main contribution of this paper is to show how stability of a hybrid system can be reduced to a specialization of the well established stability theory of differential inclusions. A number of examples illustrate the concepts introduced in the paper.

Optimal control of higher order differential inclusions with functional constraints

2020 ◽  
Vol 26 ◽  
pp. 37 ◽  
Author(s):  
Elimhan N. Mahmudov
Keyword(s):  
Optimal Control ◽  
Optimality Conditions ◽  
Differential Inclusion ◽  
Differential Inclusions ◽  
Higher Order ◽  
Second Order ◽  
Sufficient Optimality Conditions ◽  
Functional Constraints ◽  
Complementary Slackness ◽  
Complementary Slackness Conditions

The present paper studies the Mayer problem with higher order evolution differential inclusions and functional constraints of optimal control theory (PFC); to this end first we use an interesting auxiliary problem with second order discrete-time and discrete approximate inclusions (PFD). Are proved necessary and sufficient conditions incorporating the Euler–Lagrange inclusion, the Hamiltonian inclusion, the transversality and complementary slackness conditions. The basic concept of obtaining optimal conditions is locally adjoint mappings and equivalence results. Then combining these results and passing to the limit in the discrete approximations we establish new sufficient optimality conditions for second order continuous-time evolution inclusions. This approach and results make a bridge between optimal control problem with higher order differential inclusion (PFC) and constrained mathematical programming problems in finite-dimensional spaces. Formulation of the transversality and complementary slackness conditions for second order differential inclusions play a substantial role in the next investigations without which it is hardly ever possible to get any optimality conditions; consequently, these results are generalized to the problem with an arbitrary higher order differential inclusion. Furthermore, application of these results is demonstrated by solving some semilinear problem with second and third order differential inclusions.

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