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

the union of convex polyhedra in three dimensions we show that the number of vertices edges and faces of the union of k convex polyhedra in 3 space having a total of n faces is o k3 kn log k this bound is almost tight in the worst case as there exist collections of polyhedra with omega k 3 kn alpha k union complexity we also describe a rather simple randomized incremental algorithm for computing the boundary of the union in o k3 kn log k log n expected time

the linear quadratic regulator for periodic hybrid systems abstract the main objective of this paper is to characterize feedback control laws that are optimal with respect to a quadratic cost functional in the framework of linear hybrid systems undergoing time driven periodic jumps namely the so called hybrid linear quadratic regulator lqr problem the optimal solution to the hybrid lqr problem is determined both in the case of finite horizon and infinite horizon optimal control problems by introducing a hybrid periodic extension of the classic differential and difference riccati equations thus leading to the notion of monodromy riccati equation interestingly due to the periodic nature of the discrete time events the computation of the optimal feedback hinges upon the solution of a differential rather than algebraic riccati equation also in the infinite horizon case hence yielding a time varying periodic control law necessary and sufficient conditions that ensure asymptotic stability of the closed loop system are provided and discussed in detail in the case of infinite horizon optimal control problems

optimal control for hybrid systems with partitioned state space for hybrid systems where the continuous state space is partitioned by switching manifolds the discrete state changes autonomously if the continuous state hits a manifold recently we introduced a version of the minimum principle for the optimal control of such systems where necessary optimality conditions were provided for the case that a trajectory passes through the intersection of switching manifolds further we introduced an algorithm based on these conditions for computing optimal controls by varying not only the continuous state on switching manifolds based on gradient information but also the sequence of discrete states it was shown that the combinatorial complexity of former computational schemes based on the minimum principle can be avoided since not all possible discrete state sequences need to be analyzed separately in this note theoretical aspects of the algorithm and practical considerations for a successful implementation are discussed in detail further a comparison to existing algorithms for the solution of hybrid optimal control problems with autonomous switching is presented the efficiency of the proposed algorithm is demonstrated by a novel numerical example

on the global controllability of nonlinear systems fountains recurrence and applications to hamiltonian systems in this paper a form of open local accessibility for nonlinear control systems is introduced called the continuous fountain condition subject to the conditions that i the states of a system are continuous fountains and ii one of various recurrence conditions holds it is established that the system state space is globally controllable it is shown that these controllability results imply the controllability of certain subsets of the state space of hamiltonian control systems called energy slices the relations between the notion of a fountain and i local accessibility and ii the full lie algebra rank condition for control affine systems are investigated finally the results in this paper are shown to have application to hierarchical hybrid control theory in that they give conditions for a finite analytic partition to satisfy the so called in block controllability property this is illustrated with a linear mass spring system example

correction to on the hybrid optimal control problem theory and algorithms this paper discusses about correction to ldquoon the hybrid optimal control problem theory and algorithms in the above paper the formulas below should read as shown y tj 1 lim irarrinfin 1 epsii delta xi tj 1 phij 1 tj 1 tj x phij tj t fj x0 t v fj x0 t u0 t gammaj

brief paper linear quadratic switching control with switching cost we study in this paper the linear quadratic lq optimal control problem of discrete time switched systems with a constant switching cost for both finite and infinite time horizons we reduce these problems into an auxiliary problem which is an lq optimal switching control problem with a cardinality constraint on the total number of switchings based on the solution structure derived from the dynamic programming dp procedure we develop a lower bounding scheme by exploiting the monotonicity of the riccati difference equation integrating such a lower bounding scheme into a branch and bound bnb framework we offer an efficient numerical solution scheme for the lq switching control problem with switching cost

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

optimal control of switched systems based on parameterization of the switching instants this paper presents a new approach for solving optimal control problems for switched systems we focus on problems in which a prespecified sequence of active subsystems is given for such problems we need to seek both the optimal switching instants and the optimal continuous inputs in order to search for the optimal switching instants the derivatives of the optimal cost with respect to the switching instants need to be known the most important contribution of the paper is a method which first transcribes an optimal control problem into an equivalent problem parameterized by the switching instants and then obtains the values of the derivatives based on the solution of a two point boundary value differential algebraic equation formed by the state costate stationarity equations the boundary and continuity conditions along with their differentiations this method is applied to general switched linear quadratic problems and an efficient method based on the solution of an initial value ordinary differential equation is developed an extension of the method is also applied to problems with internally forced switching examples are shown to illustrate the results in the paper

modal occupation measures and lmi relaxations for nonlinear switched systems control this paper presents a linear programming approach for the optimal control of nonlinear switched systems where the control is the switching sequence this is done by introducing modal occupation measures which allow to relax the problem as a primal linear programming lp problem its dual linear program of hamilton jacobi bellman inequalities is also characterized the lps are then solved numerically with a converging hierarchy of primal dual moment sum of squares sos linear matrix inequalities lmi because of the special structure of switched systems we obtain a much more efficient method than could be achieved by applying standard moment sos lmi hierarchies for general optimal control problems

direct methods with maximal lower bound for mixed integer optimal control problems many practical optimal control problems include discrete decisions these may be either time independent parameters or time dependent control functions as gears or valves that can only take discrete values at any given time while great progress has been achieved in the solution of optimization problems involving integer variables in particular mixed integer linear programs as well as in continuous optimal control problems the combination of the two is yet an open field of research we consider the question of lower bounds that can be obtained by a relaxation of the integer requirements for general nonlinear mixed integer programs such lower bounds typically suffer from a huge integer gap we convexify with respect to binary controls and relax the original problem and prove that the optimal solution of this continuous control problem yields the best lower bound for the nonlinear integer problem building on this theoretical result we present a novel algorithm to solve mixed integer optimal control problems with a focus on discrete valued control functions our algorithm is based on the direct multiple shooting method an adaptive refinement of the underlying control discretization grid and tailored heuristic integer methods its applicability is shown by a challenging application the energy optimal control of a subway train with discrete gears and velocity limits

an optimal algorithm for intersecting three dimensional convex polyhedra this paper describes a linear time algorithm for computing the intersection of two convex polyhedra in 3 space applications of this result to computing intersections convex hulls and voronoi diagrams are also given

optimal control of switching systems this paper considers an optimal control problem for a switching system for solving this problem we do not make any assumptions about the number of switches nor about the mode sequence they are determined by the solution of the problem the switching system is embedded into a larger family of systems and the optimization problem is formulated for the latter it is shown that the set of trajectories of the switching system is dense in the set of trajectories of the embedded system the relationship between the two sets of trajectories 1 motivates the shift of focus from the original problem to the more general one and 2 underlies the engineering relevance of the study of the second problem sufficient and necessary conditions for optimality are formulated for the second optimization problem if they exist bang bang type solutions of the embedded optimal control problem are solutions of the original problem otherwise suboptimal solutions are obtained via the chattering lemma

a switched lq regulator design in continuous time in this technical note the design of a lq regulator for linear switched systems in continuous time is investigated from a relaxation of the optimal control problem a lyapunov based switching law is provided even if the subsystems are all unstable the state feedback switching law can be applied subject to a positiveness condition in any cases the real cost is always upper bounded by the lyapunov function value the optimality of the switching law is also discussed and we prove that the switching conditions are optimal in some generic cases this point explains why the obtained results over examples approach finely the optimal solutions finally a design strategy is also given that extends the results to the cases where the subsystems are controlled linear systems

on minimum time paths of bounded curvature with position dependent constraints we consider the problem of a particle traveling from an initial configuration to a final configuration given by a point in the plane along with a prescribed velocity vector in minimum time with non homogeneous velocity and with constraints on the minimum turning radius of the particle over multiple regions of the state space necessary conditions for optimality of these paths are derived to characterize the nature of optimal paths both when the particle is inside a region and when it crosses boundaries between neighboring regions these conditions are used to characterize families of optimal and nonoptimal paths among the optimality conditions we derive a refraction law at the boundary of the regions that generalizes the so called snell s law of refraction in optics to the case of paths with bounded curvature tools employed to deduce our results include recent principles of optimality for hybrid systems a numerical example is given to demonstrate the derived results

applications of optimal multiprocesses in a previous paper a theory of necessary conditions for optimal multiprocess problems was supplied that is problems in dynamic optimization involving a collection of control systems coupled through constraints on the endpoints of the trajectories and the cost functions here the scope of the new theory is illustrated by application to optimization problems in investment planning impulse control optics robotics and renewable resources

quadratic optimal control of linear complementarity systems first order necessary conditions and numerical analysis this article is dedicated to the analysis of quadratic optimal control of linear complementarity systems lcss which are a class of strongly nonlinear and nonsmooth dynamical systems necessary first order conditions are derived that take the form of an lcs with inequality constraints hence are numerically tractable sufficiency is proved in a specific sense numerical examples illustrate the theoretical developments and demonstrate the efficiency of the complementarity approach

fat triangles determine linearly many holes the authors show that for every fixed delta 0 the following holds if f is a union of n triangles all of whose angles are at least delta then the complement of f has o n connected components and the boundary of f consists of o n log log n straight segments where the constants of proportionality depend on delta this latter complexity becomes linear if all triangles are of roughly the same size or if they are all infinite wedges

computational geometry an introduction from the reviews this book offers a coherent treatment at the graduate textbook level of the field that has come to be known in the last decade or so as computational geometry the book is well organized and lucidly written a timely contribution by two founders of the field it clearly demonstrates that computational geometry in the plane is now a fairly well understood branch of computer science and mathematics it also points the way to the solution of the more challenging problems in dimensions higher than two

control of admission and routing in loss networks hybrid dynamic programming equations call admission and routing control decisions in stochastic loss circuit switched networks with semi markovian multi class call arrival and general connection time processes are formulated as optimal stochastic control problems the resulting so called hybrid dynamic programming equation systems take the form of vectors of partial differential equations with each component associated to a distinct distribution of routed calls over the network i e distinct occupation states this framework reduces to that of a markov decision process when the traffic is poisson and the associated computational limitations are approximately those of linear programs examples are provided of i network state space constructions and controlled state transition processes ii a new closed form solution for a simple network and iii the analysis and illustrative numerical results for a three link network

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

a simple algorithm for principalization of monomial ideals in this paper we give a simple constructive proof of principalization of monomial ideals and the global analog this also gives an algorithm for principalization

a hamiltonian approach to compute an energy efficient trajectory for a servomotor system this paper considers a nonlinear constrained optimal control problem ncocp originated from energy optimal trajectory planning of servomotor systems solving the exact optimal solution is challenging because of the nonlinear and switching cost function and various constraints this paper proposes a method to manage the switching cost function to establish a set of necessary conditions of an ncocp specifically a concept sub trajectory is introduced to match multiple hamiltonian due to switches in the cost function necessary conditions on the optimal trajectory are established as a union of conditions for all sub trajectories and weierstrass erdmann corner conditions between sub trajectories the set of feasible structures of optimal trajectories is further identified and represented by various state transition diagrams for the servomotor application a decomposition based shooting method is proposed to compute an optimal trajectory by solving multi point boundary value problems simulations and experiments validate the effectiveness of the methodology and the energy saving benefit

optimal linear quadratic regulator of switched systems this paper considers the optimal control problem of linear switched systems with linear quadratic lq cost or multiple lq cost by adopting an embedding transformation the switching design problem is relaxed and transformed into a traditional optimal control problem the bang bang type solutions of the embedded optimal control problems are obtained for both the positive definite lq cost case and the multiple lq cost case which are the optimal solutions to the original problems the switching sequence of modes and the switching instants can be calculated by solving a closed form optimal switching condition the optimal state feedback control law is determined simultaneously finally numerical results are provided to illustrate the effectiveness of the proposed method

on optimal control problems of a class of impulsive switching systems with terminal states constraints abstract the global optimal control problem is proposed for a special class of hybrid dynamical systems i e impulsive switching systems then the necessary condition of the above problem the minimum principle is given ekeland s variational principle and the matrix cost functional structure expression are utilized in the process of the proof based on the main result a special linear hybrid impulsive and switching system hiss is illustrated and the optimal control algorithm is presented moreover the cases of pure impulsive systems and pure switched systems are included in this paper

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

correspondence comments on optimally switched linear systems the technical note titled optimally switched linear systems by the authors t das and r mukherjee addresses the problem of optimal switching for switched systems the proof of the theorem in the paper is incorrect and point out an important theoretical point which explains why the optimal switched solution does not exist whenever there exists a singular one to a relaxed problem this point has also important consequences concerning numerical difficulties encountered in solving such problems and are also discussed

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