continuous representation and control of hybrid systems this article introduces a continuous representation for controlled model and state jump hybrid systems such a representation is valuable in that it manages to cope with the difficulties related to discontinuities by replacing the study of these systems by that of continuous ones which is simple the studied hybrid systems are first introduced together with their associated continuous systems some localized and global convergences of the states of continuous systems to those of hybrid systems are proved then the continuous representation is used within the optimal control framework free end time and trajectory pursuit hybrid problems are studied as well as their respective continuous versions convergences of solutions between continuous and hybrid problems are established which shows that the resolution of a problem ensuing from the representation enables one to obtain an approximation of the solution of the initial hybrid problem finally the formalism of calculus of variations is used to obta

convergence results for smooth regularizations of hybrid nonlinear optimal control problems we consider a class of hybrid nonlinear optimal control problems having a discontinuous dynamics ruled by a partition of the state space for this class of problems some hybrid versions of the usual pontryagin maximum principle are known we introduce general regularization procedures parameterized by a small parameter smoothing the previous hybrid problems to standard smooth optimal control problems for which we can apply the usual pontryagin maximum principle we investigate the question of the convergence of the resulting extremals as the regularization parameter tends to zero under some general assumptions we prove that smoothing regularization procedures converge in the sense that the solution of the regularized problem as well as its extremal lift converges to the solution of the initial hybrid problem to illustrate our convergence result we apply our approach to the minimal time low thrust coplanar orbit transfer with eclipse constraint

stability analysis and control of rigid body systems with impacts and friction many critical tasks in robotics such as locomotion or manipulation involve collisions between a rigid body and the environment or between multiple bodies methods based on sums of squares sos for numerical computation of lyapunov certificates are a powerful tool for analyzing the stability of continuous nonlinear systems and can additionally be used to automatically synthesize stabilizing feedback controllers here we present a method for applying sums of squares verification to rigid bodies with coulomb friction undergoing discontinuous inelastic impact events the proposed algorithm explicitly generates lyapunov certificates for stability positive invariance and safety over admissible non penetrating states and contact forces we leverage the complementarity formulation of contact which naturally generates the semialgebraic constraints that define this admissible region the approach is demonstrated on multiple robotics examples including simple models of a walking robot a perching aircraft and control design of a balancing robot

bisimulation relations for dynamical control and hybrid systems the fundamental notion of bisimulation equivalence for concurrent processes has escaped the world of continuous and subsequently hybrid systems inspired by the categorical framework of joyal nielsen and winskel we develop novel notions of bisimulation equivalence for dynamical systems as well as control systems we prove that these notions can be captured by the abstract notion of bisimulation as developed by joyal nielsen and winskel this is the first unified notion of system equivalence that transcends discrete and continuous systems furthermore this enables the development of a novel and natural notion of bisimulation for hybrid systems which is the final goal of this paper

optimal control for pointwise asymptotic stability in a hybrid control system abstract pointwise asymptotic stability or semistability is a property of the set of equilibria of a dynamical system where every equilibrium is lyapunov stable and every solution is convergent to some equilibrium under an appropriate version of asymptotic controllability assumption it is shown that the property can be achieved in a hybrid control system by open loop optimal solutions of an infinite horizon optimal control problem for discrete time systems the optimal solutions can be generated by feedback regularity of the optimal value function and the existence of hybrid optimal controls are also studied

supervisory control of hybrid systems under partial observation based on l complete approximations this paper addresses a supervisory control problem for time invariant hybrid dynamical systems based on the l complete approximation scheme by moor and raisch discrete supervisory control of hybrid systems based on l complete approximations 2002 and partial observation an underlying hybrid plant is realized by a hybrid state machine with an infinite state space and its external behavior is described by discrete input output signals with finite range and some output signal is assumed to be unmeasurable for the strongest l complete approximation of the plant which is realized by a finite state machine we present a method to design a finite state supervisor to achieve a given specification the supervisor cannot observe the unmeasurable output signal finally we show that the finite state supervisor also meets the specification for the underlying hybrid plant

controllability and observability for a class of controlled switching impulsive systems in this note we study the controllability and observability problem for a class of controlled switching impulsive systems by proposing several formula of variation of parameters for this type of time varying systems and employing the characteristic polynomial theory of matrix we establish necessary and sufficient conditions for controllability and controlled observability with respect to a given switching time sequence specializing the obtained results to the case of time invariant linear switching impulsive systems we derive some simple algebraic criteria which include the results reported in the literature for time invariant linear switching systems linear impulsive systems and classical linear systems one example is worked out for illustration

technical communique optimally switched linear systems in this paper we address the problem of optimal switching for switched linear systems the uniqueness of our approach lies in describing the switching action by multiple control inputs this allows us to embed the switched system in a larger family of systems and apply pontryagin s minimum principle for solving the optimal control problem this approach imposes no restriction on the switching sequence or the number of switchings this is in contrast to search based algorithms where a fixed number of switchings is set a priori in our approach the optimal solution can be determined by solving the ensuing two point boundary value problem results of numerical simulations are provided to support the proposed method

breather solutions of the discrete nonlinear schrodinger equations with unbounded potentials in this paper i investigate the existence of nontrivial breather solutions of the discrete nonlinear schrodinger equation with the unbounded potential at infinity first i derive a discrete version of compact embedding theorem then combining the nehari manifold approach and the compact embedding theorem i show the existence of breather solutions without palais smale condition the results on the exponential decay of breather solutions are also provided in this paper

optimal control of cell mass and maturity in a model of follicular ovulation in this paper we study optimal control problems associated with a scalar hyperbolic conservation law modeling the development of ovarian follicles changes in the age and maturity of follicular cells are described by a two dimensional conservation law where the control terms act on the velocities the control problem consists in optimizing the follicular cell resources so that the follicular maturity reaches a maximal value in fixed time formulating the optimal control problem within a hybrid framework we prove necessary optimality conditions in the form of the hybrid maximum principle then we derive the optimal strategy and show that there exists at least one optimal bang bang control with one single switching time

a numerical algorithm for optimal control of a class of hybrid systems differential transformation based approach a novel numerical algorithm based on differential transformation is proposed for optimal control of a class of hybrid systems with a predefined mode sequence from the necessary conditions for optimality of hybrid systems the hybrid optimal control problem is first converted into a two point boundary value problem tpbvp with additional transverse conditions at the switching times then we propose a differential transformation algorithm for solving the tpbvp which may have discontinuities in the state and or control input at the switching times using differential transformation the hybrid optimal control problem reduces to a problem of solving a system of algebraic equations the numerical solution is obtained in the form of a truncated taylor series by taking advantage of the special properties of the linear subsystems and a quadratic cost functional the differential transformation algorithm can be further simplified for the switched linear quadratic optimal control problem we analyse the error of

some combinatorial properties of jack symmetric functions if 1 l n then write 2 n or 1 n if p is another partition then write p c j if pi 6 for all i i e if the diagram of 1 contains the diagram of p if ipl ii 1 then write p 2 1 reverse lexicographic order if either j i or the first nonvanishing difference e pi is positive for instance writing 2 1 i for ai i it 5 5 41 32 5 311 221 2111 11111 finally write p i if ipl lal and p1 pl2 pli a1 a2 li for all i macdonald m p 63 call the partial ordering the natural ordering but we will call it the dominance ordering let x x x2 be an infinite set of indeterminates as in mi we

brief on solution concepts and well posedness of linear relay systems in this paper we study the well posedness existence and uniqueness of solutions of linear relay systems with respect to two different solution concepts filippov solutions and forward solutions we derive necessary and sufficient conditions for well posedness in the sense of filippov of linear systems of relative degree one and two in closed loop with relay feedback to be precise uniqueness of filippov and also forward solutions follows in this case if the first non zero markov parameter is positive by means of an example it is shown that this intuitively clear condition is not true for systems with relative degree larger than two the influence of the zeno phenomenon an infinite number of relay switching times in a finite length time interval on well posedness is highlighted and although linear relay systems form a rather limited subclass of hybrid dynamical systems the consequences of the presence of the zeno behaviour is typical for many other classes of non smooth and hybrid systems

hierarchical trajectory optimization for a class of hybrid dynamical systems we discuss trajectory optimization for a class of hybrid systems with a natural hierarchical separation of discrete and continuous dynamics where the continuous dynamics do not change with the discrete state the trajectory optimization problem considered requires that a discrete state sequence and a continuous state trajectory must be both determined to minimize a single cost function such that the discrete state sequence also solves a symbolic planning problem we model this symbolic planning problem as a search on a planning graph and we introduce a family of graphs called lifted planning graphs parametrized by an integer h we define a family of continuous state trajectory optimization problems and associate them with edge costs in the lifted planning graphs next we present an algorithm for finding an optimal solution to the hybrid trajectory optimization problem which includes mapping paths in the lifted planning graphs to discrete state sequences and continuous state trajectories we show that the cost of optimal hybrid trajectories is a nonincreasing function of h and that there exists a finite h for which this cost attains a minimum we illustrate the proposed algorithm with numerical simulation results for two application examples an autonomous mobile vehicle and an autonomous robotic manipulator

the hybrid maximum principle is a consequence of pontryagin maximum principle we give a simple proof of the maximum principle for smooth hybrid control systems by reducing the hybrid problem to an optimal control problem of pontryagin type and then by using the classical pontryagin maximum principle

supervisory control of hybrid systems within a behavioural framework abstract this contribution addresses the synthesis of supervisory control for hybrid systems with discrete external signals such systems are in general neither l complete nor can they be represented by finite state machines we find an l complete approximation abstraction l for represent it by a finite state machine and investigate the control problem for the approximation if a solution exists we synthesize the maximally permissive supervisor for l we show that it also solves the control problem for the hybrid system if no solution exists approximation accuracy can be increased by computing a k complete abstraction k k l this paper is entirely set within the framework on willems behavioural systems theory

control of quantized systems based on discrete event models this paper presents controller design methods for dynamical systems that are observed by discrete sensors these sensors induce a partitioning of the state space and only this quantized information is available for the controller the so called quantized system is modelled by a discrete event model that serves as a basis for the controller design methods however instead of using solely the classical control methodologies for discrete event systems as found in the literature improvements are proposed by including additional information provided by the fact that the underlying plant is continuous by nature such as continuity of the state trajectories and information on derivatives that holds for parts of the state space the concept of discretely controlled invariant sets will play a crucial role in the development of control strategies and necessary and sufficient conditions for controlled invariance are presented also algorithms are included to compute the smallest and largest discretely controlled

differential automata and their discrete simulators on presente la notion d automate differentiel pour la formulation d une classe de problemes aux valeurs initiales sur des champs vectoriels hysteretiques

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

smoothing property for schrodinger equations with potential superquadratic at infinity we prove a smoothing property for one dimensional time dependent schrodinger equations with potentials which satisfy at infinity k 2 as an application we show that the initial value problem for certain nonlinear schrodinger equations with such potentials is l2 well posed we also prove a sharp asymptotic estimate of the lp norm of the normalized eigenfunctions of h v for large energy

quasi optimal robust stabilization of control systems in this paper we investigate the problem of semiglobal minimal time robust stabilization of analytic control systems with controls entering linearly by means of a hybrid state feedback law it is shown that in the absence of minimal time singular trajectories the solutions of the closed loop system converge to the origin in quasi minimal time for a given bound on the controller with a robustness property with respect to small measurement noise external disturbances and actuator noise

breather solutions of the discrete nonlinear schrodinger equations with sign changing nonlinearitya in this paper we investigate the breather solutions of the discrete nonlinear schrodinger equation with the unbounded potential and nonlinearity containing both self focusing and defocusing sites which means the coefficients of nonlinearity change sign by using the generalized nehari manifold method we obtain the existence of nontrivial exponential decay breather solutions

breathers in nonlinear lattices existence linear stability and quantization abstract it has been proved by the principle of anti integrable limit renamed here anti continuous limit and the implicit function theorem that under rather weak hypotheses there exist spatially localized and time periodic solutions breathers in arrays of nonlinear coupled classical oscillators provided the coupling is not too strong the models can be translationally invariant or not and be of any dimension there are also time periodic solutions corresponding to arbitrary distributions on the lattice of such solutions multibreathers the whole set of these solutions can be labeled by a discrete coding sequence the condition that the frequency of the breather or multibreather solution has no harmonics in the phonon spectrum can be discarded for a large subset of spatially extended time periodic solutions called phonobreather which correspond to special codes which can be also spatially chaotic and where no oscillator is at rest these results extend almost without change to models with coupled rotors the existence of rotobreathers which are exact solutions consisting of a single rotor rotating while the other rotors oscillate is proved there are also multirotobreather solutions and phono rotobreathers etc it is also shown that the phase of the single breathers constituting a multibreather or multirotobreather solution can be submitted to a phase torsion these new dynamical structures carry a nonzero energy flow in models with two dimensions and more the phase torsion of multibreather solutions can generate new dynamical solutions with vortices in the energy flow the dynamics of a quantum electron coupled to a classical lattice is also studied within the same approach from the anti continuous limit the existence of two kinds of polarobreathers which are spatially exponentially localized dynamical solutions is proved for a small enough electronic transfer integral the polarobreathers of the first kind include the standard static polaron the lattice configuration is static time independent when the electron is in an excited state i e the electronic oscillator oscillates periodically they can be spatially chaotic a polarobreather of the second kind corresponds to a localized periodic lattice oscillation associated with a localized quasiperiodic electronic oscillation the most simple of these solutions are those for which both the lattice oscillation and the electron wave function are mostly concentrated close to the same single site the standard floquet analysis of the linear stability of the breather and multibreather solutions is shown to be related with the band spectrum property of the newton operator involved in the implicit function theorem the krein theory of bifurcations is reinterpreted from this point of view the linear stability of the single breather and rotobreather solutions is then proved at least at small enough coupling finally it is shown that in the quantum version of translationally invariant classical models where there cannot exist in principle any strictly localized excitation there are narrow bands of excitations consisting of bound states of many bosons which are the quantum counterpart of the classical breathers the bandwidth of these excitations goes exponentially to zero with the number of bound phonons this theory provides simultaneously existence theorems and a new powerful and accurate numerical method for calculating practically any of the solutions which has been formally predicted to exist these numerical applications are currently written elsewhere

toward unified analysis and controller synthesis for a class of hybrid systems abstract this work defines a new class of hybrid systems called state based switched sbs systems that have numerous important engineering applications the characterizing feature of these systems is that the discrete event dynamics are associated with the continuous time state making a specific function be equal to zero the choice of this function is application specific and for the closed loop sbs systems defined in this paper it is related to the execution of a desired set of tasks from a pre specified mission plan for this broad class of sbs systems the paper presents a unified analysis and controller synthesis methodology based on lyapunov theory depending on the details of the mission plan the closed loop hybrid system will be divided into two subclasses sequential and non sequential the controller design procedure for both subclasses consists of the same two steps finding a control law and finding a stabilizing switching rule for static state and output feedback of sequential hybrid systems the paper proposes a new hybrid sequential sliding mode controller it is proven that the control mission can be accomplished for sequential hybrid systems under static state and output feedback using this new controller a similar framework is investigated for the more complex class of nonsequential hybrid systems and a systematic procedure for designing the switching rule is presented for some specific instances of these systems

form factors for quasi particles in c 1 conformal field theory the non fermi liquid physics at the edge of fractional quantum hall systems is described by specific chiral conformal field theories with central charge c 1 the charged quasi particles in these theories have fractional charge and obey a form of fractional statistics in this paper we study form factors which are matrix elements of physical conformal operators evaluated in a quasi particle basis that is organized according to the rules of fractional exclusion statistics using the systematics of jack polynomials we derive selection rules for a special class of form factors we argue that finite temperature green functions can be evaluated via systematic form factor expansions using form factors such as those computed in this paper and thermodynamic distribution functions for fractional exclusion statistics we present a specific case study where we demonstrate that the form factor expansion shows a rapid convergence

a form factor approach to finite temperature correlation functions in c 1 cft the excitation spectrum of specific conformal field theories cft with central charge c 1 can be described in terms of quasi particles with charges q p 1 and fractional statistics properties using the language of jack polynomials we compute form factors of the charge density operator in these cfts we study a form factor expansion for the finite temperature density density correlation function and find that it shows a quick convergence to the exact result the low temperature behaviour is recovered from a form factor with p 1 particles while the high temperature limit is recovered from states containing no more than three particles

convex dynamic programming for hybrid systems a classical linear programming approach to optimization of flow or transportation in a discrete graph is extended to hybrid systems the problem is finite dimensional if the state space is discrete and finite but becomes infinite dimensional for a continuous or hybrid state space it is shown how strict lower bounds on the optimal loss function can be computed by gridding the continuous state space and restricting the linear program to a finite dimensional subspace upper bounds can be obtained by evaluation of the corresponding control laws

the hierarchical lattices of a finite machine the concept of dynamical consistency is introduced to define a high level transition function for an arbitrary partition of the state space of a finite input state machine m the existence of a dynamically consistent transition from a partition member xi to a member xj corresponds to the existence for every state x in xi of some x dependent control sequence which takes that state directly into xj this paper analyses what are termed in block controllable partition machines and between block controllable partition machines and establishes the existence of the associated in block controllable hierarchical lattice ibcp m it is shown that any partition machine in ibcp m is between block controllable if and only if the base machine m is controllable this setting allows the specification of state to state control trajectories of m to be decomposed into high level and low level components

optimal control of a class of hybrid systems we present a modeling framework for hybrid systems intended to capture the interaction of event driven and time driven dynamics this is motivated by the structure of many manufacturing environments where discrete entities termed jobs are processed through a network of workcenters so as to change their physical characteristics associated with each job is a temporal state subject to event driven dynamics and a physical state subject to time driven dynamics based on this framework we formulate and analyze a class of optimal control problems for single stage processes first order optimality conditions are derived and several properties of optimal state trajectories sample paths are identified which significantly simplify the task of obtaining explicit optimal control policies

local smoothing property and strichartz inequality for schrodinger equations with potentials superquadratic at infinity abstract we study smoothing properties for time dependent schrodinger equations i u t 1 2 u v x u u 0 l 2 r n with potentials which satisfy v x o x m at infinity m 2 we show that the solution u t x is 1 m times differentiable with respect to x at almost all t r and explain that this is the result of the fact that the sojourn time of classical particles with energy in arbitrary compact set is less than ct 1 m during 0 t when is very large we also show strichartz s inequality with derivative loss for such potentials and give its application to nonlinear schrodinger equations

on the quasiparticle description of c 1 cfts we show that the description of c 1 conformal field theory in terms of quasiparticles satisfying fractional statistics can be obtained from the sine gordon model with a chemical potential a in the limit where a m these quasiparticles are related to the excitations of the calogero sutherland cs model we provide a direct calculation of their two particle s matrix using korepin s method we also reconsider the computation of the cs s matrix in terms of particles with fractional charge

brief energy based control for hybrid port controlled hamiltonian systems in this paper we develop an energy based hybrid control framework for hybrid port controlled hamiltonian systems in particular we obtain constructive sufficient conditions for hybrid feedback stabilization that provide a shaped energy function for the closed loop system while preserving a hybrid hamiltonian structure at the closed loop level furthermore an inverse optimal hybrid feedback control framework is developed that characterizes a class of globally stabilizing energy based controllers that guarantee hybrid sector and gain margins to multiplicative input uncertainty of hybrid hamiltonian systems

global existence and compact attractors for the discrete nonlinear schrodinger equation abstract we study the asymptotic behavior of solutions of discrete nonlinear schrodinger type dnls equations for a conservative system we consider the global in time solvability and the question of existence of standing wave solutions similarities and differences with the continuous counterpart nls partial differential equation are pointed out for a dissipative system we prove existence of a global attractor and its stability under finite dimensional approximations similar questions are treated in a weighted phase space finally we propose possible extensions for various types of dnls equations

the rational complementarity problem abstract an extension of the linear complementarity problem lcp of mathematical programming is the so called rational complementarity problem rcp this problem occurs if complementarity conditions are imposed on input and output variables of linear dynamical input state output systems the resulting dynamical systems are called linear complementarity systems since the rcp is crucial both in issues concerning existence and uniqueness of solutions to complementarity systems and in time simulation of complementarity systems it is worthwhile to consider existence and uniqueness questions of solutions to the rcp in this paper necessary and sufficient conditions are presented guaranteeing existence and uniqueness of solutions to the rcp in terms of corresponding lcps using these results and proving that the corresponding lcps have certain properties we can show uniqueness and existence of solutions to linear mechanical systems with unilateral constraints electrical networks with diodes and linear dynamical systems subject to relays and or coulomb friction

hybrid necessary principle we consider a hybrid control system and general optimal control problems for this system we suppose that the switching strategy imposes restrictions on control sets and we provide necessary conditions for an optimal hybrid trajectory stating a hybrid necessary principle hnp our result generalizes various necessary principles available in the literature

a unified framework for hybrid control model and optimal control theory we propose a very general framework that systematizes the notion of a hybrid system combining differential equations and automata governed by a hybrid controller that issues continuous variable commands and makes logical decisions we first identify the phenomena that arise in real world hybrid systems then we introduce a mathematical model of hybrid systems as interacting collections of dynamical systems evolving on continuous variable state spaces and subject to continuous controls and discrete transitions the model captures the identified phenomena subsumes previous models yet retains enough structure to pose and solve meaningful control problems we develop a theory for synthesizing hybrid controllers for hybrid plants in all optimal control framework in particular we demonstrate the existence of optimal relaxed and near optimal precise controls and derive generalized quasi variational inequalities that the associated value function satisfies we summarize algorithms for solving these inequalities based on a generalized bellman equation impulse control and linear programming

universal computation and other capabilities of hybrid and continuous dynamical systems we explore the simulation and computational capabilities of hybrid and continuous dynamical systems the continuous dynamical systems considered are ordinary differential equations odes for hybrid systems we concentrate on models that combine odes and discrete dynamics e g finite automata we review and compare four such models from the literature notions of simulation of a discrete dynamical system by a continuous one are developed we show that hybrid systems whose equations can describe a precise binary timing pulse exact clock can simulate arbitrary reversible discrete dynamical systems defined on closed subsets of r n the simulations require continuous odes in ir2n with the exact clock as input all four hybrid systems models studied here can implement exact clocks we also prove that any discrete dynamical system in rn can be simulated by continuous odes in rt2n 1 we use this to show that smooth odes in ri3 can simulate arbitrary turing machines and hence possess the power of universal computation we use the famous asynchronous arbiter problem to distinguish between hybrid and continuous dynamical systems we prove that one cannot build an arbiter with devices described by a system of lipschitz odes on the other hand all four hybrid systems models considered can implement arbiters even if their odes are lipschitz

hybridization of optimal control problems optimization problems for hybrid systems have attracted a lot of attention in recent years this interest stimulated numerous scientific works and the development of tools to study optimal trajectories such as necessary conditions the idea of the present contribution is to use such results for hybrid systems to construct a hybridization of an optimal control problem the approximation via a hybrid system was already considered in the literature but using different methods to obtain a reduction to minimal path problems on a graph our procedure gives an efficient approximation i e more efficient than standard discretization

impulse differential inclusions a viability approach to hybrid systems impulse differential inclusions are introduced as a framework for modeling hybrid phenomena connections to standard problems in the area of hybrid systems are discussed conditions are derived that allow one to determine whether a set of states is viable or invariant under the action of an impulse differential inclusion for sets that violate these conditions methods are developed for approximating their viability and invariance kernels that is the largest subset that is viable or invariant under the action of the impulse differential inclusion the results are demonstrated on examples

level set approach for reachability analysis of hybrid systems under lag constraints this study aims at characterizing a reachable set of a hybrid dynamical system with a lag constraint in the switch control the setting does not consider any controllability assumptions and uses a level set approach the approach consists of the introduction of an adequate hybrid optimal control problem with lag constraints on the switch control whose value function allows a characterization of the reachable set the value function is in turn characterized by a system of quasi variational inequalities sqvi we prove a comparison principle for the sqvi which shows uniqueness of its solution a class of numerical finite difference scheme for solving the system of inequalities is proposed and the convergence of the numerical solution toward the value function is studied using the comparison principle some numerical examples illustrating the method are presented our study is motivated by an industrial application we are interested in the maximum range of hybrid vehicles

on the hybrid optimal control problem theory and algorithms a class of hybrid optimal control problems hocp for systems with controlled and autonomous location transitions is formulated and a set of necessary conditions for hybrid system trajectory optimality is presented which together constitute generalizations of the standard maximum principle these are given for the cases of open bounded control value sets and compact control value sets the derivations in the paper employ i classical variational and needle variation techniques and ii a local controllability condition which is used to establish the adjoint and hamiltonian jump conditions in the autonomous switching case employing the hybrid minimum principle hmp necessary conditions a class of general hmp based algorithms for hybrid systems optimization are presented and analyzed for the autonomous switchings case and the controlled switchings case using results from the theory of penalty function methods and ekeland s variational principle the convergence of these algorithms is established under reasonable assumptions the efficacy of the proposed algorithms is illustrated via computational examples

stabilizability of switched linear systems does not imply the existence of convex lyapunov functions counterexamples are given which show that a linear switched system with controlled switching that can be stabilized by means of a suitable switching law does not necessarily admit a convex lyapunov function both continuous and discrete time cases are considered this fact contributes in focusing the difficulties encountered so far in the theory of stabilization of switched system in particular the result is in contrast with the case of uncontrolled switching in which it is known that if a system is stable under arbitrary switching then admits a polyhedral norm as a lyapunov function

brief paper on optimal control of non autonomous switched systems with a fixed mode sequence we consider differentiability with respect to the switch times of the value function of an optimal control problem for a non autonomous switched system the control variables are the switch times between the modes and the input in each mode we provide a method to compute the derivative of the cost function given a nominal input then we view the optimal control problem as a parametrized optimization problem in which the switch times are the parameters and the optimization is over the set of feasible inputs of each mode from this point of view we provide conditions under which the continuity and differentiability of the optimal value function that is the cost function optimized over the inputs can be guaranteed