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.

On the Hybrid Optimal Control Problem: Theory and Algorithms

This article surveys the usual techniques of nonlinear optimal control such as the Pontryagin Maximum Principle and the conjugate point theory, and how they can be implemented numerically, with a special focus on applications to aerospace problems. In practice the knowledge resulting from the maximum principle is often insufficient for solving the problem, in particular because of the well-known problem of initializing adequately the shooting method. In this survey article it is explained how the usual tools of optimal control can be combined with other mathematical techniques to improve significantly their performances and widen their domain of application. The focus is put onto three important issues. The first is geometric optimal control, which is a theory that has emerged in the 1980s and is combining optimal control with various concepts of differential geometry, the ultimate objective being to derive optimal synthesis results for general classes of control systems. Its applicability and relevance is demonstrated on the problem of atmospheric reentry of a space shuttle. The second is the powerful continuation or homotopy method, consisting of deforming continuously a problem toward a simpler one and then of solving a series of parameterized problems to end up with the solution of the initial problem. After having recalled its mathematical foundations, it is shown how to combine successfully this method with the shooting method on several aerospace problems such as the orbit transfer problem. The third one consists of concepts of dynamical system theory, providing evidence of nice properties of the celestial dynamics that are of great interest for future mission design such as low-cost interplanetary space missions. The article ends with open problems and perspectives.

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Optimal Control and Applications to Aerospace: Some Results and Challenges

Preface Prologue: The Exponential Function Chapter 1: Abstract Integration Set-theoretic notations and terminology The concept of measurability Simple functions Elementary properties of measures Arithmetic in [0, ] Integration of positive functions Integration of complex functions The role played by sets of measure zero Exercises Chapter 2: Positive Borel Measures Vector spaces Topological preliminaries The Riesz representation theorem Regularity properties of Borel measures Lebesgue measure Continuity properties of measurable functions Exercises Chapter 3: Lp-Spaces Convex functions and inequalities The Lp-spaces Approximation by continuous functions Exercises Chapter 4: Elementary Hilbert Space Theory Inner products and linear functionals Orthonormal sets Trigonometric series Exercises Chapter 5: Examples of Banach Space Techniques Banach spaces Consequences of Baire's theorem Fourier series of continuous functions Fourier coefficients of L1-functions The Hahn-Banach theorem An abstract approach to the Poisson integral Exercises Chapter 6: Complex Measures Total variation Absolute continuity Consequences of the Radon-Nikodym theorem Bounded linear functionals on Lp The Riesz representation theorem Exercises Chapter 7: Differentiation Derivatives of measures The fundamental theorem of Calculus Differentiable transformations Exercises Chapter 8: Integration on Product Spaces Measurability on cartesian products Product measures The Fubini theorem Completion of product measures Convolutions Distribution functions Exercises Chapter 9: Fourier Transforms Formal properties The inversion theorem The Plancherel theorem The Banach algebra L1 Exercises Chapter 10: Elementary Properties of Holomorphic Functions Complex differentiation Integration over paths The local Cauchy theorem The power series representation The open mapping theorem The global Cauchy theorem The calculus of residues Exercises Chapter 11: Harmonic Functions The Cauchy-Riemann equations The Poisson integral The mean value property Boundary behavior of Poisson integrals Representation theorems Exercises Chapter 12: The Maximum Modulus Principle Introduction The Schwarz lemma The Phragmen-Lindelof method An interpolation theorem A converse of the maximum modulus theorem Exercises Chapter 13: Approximation by Rational Functions Preparation Runge's theorem The Mittag-Leffler theorem Simply connected regions Exercises Chapter 14: Conformal Mapping Preservation of angles Linear fractional transformations Normal families The Riemann mapping theorem The class L Continuity at the boundary Conformal mapping of an annulus Exercises Chapter 15: Zeros of Holomorphic Functions Infinite Products The Weierstrass factorization theorem An interpolation problem Jensen's formula Blaschke products The Muntz-Szas theorem Exercises Chapter 16: Analytic Continuation Regular points and singular points Continuation along curves The monodromy theorem Construction of a modular function The Picard theorem Exercises Chapter 17: Hp-Spaces Subharmonic functions The spaces Hp and N The theorem of F. and M. Riesz Factorization theorems The shift operator Conjugate functions Exercises Chapter 18: Elementary Theory of Banach Algebras Introduction The invertible elements Ideals and homomorphisms Applications Exercises Chapter 19: Holomorphic Fourier Transforms Introduction Two theorems of Paley and Wiener Quasi-analytic classes The Denjoy-Carleman theorem Exercises Chapter 20: Uniform Approximation by Polynomials Introduction Some lemmas Mergelyan's theorem Exercises Appendix: Hausdorff's Maximality Theorem Notes and Comments Bibliography List of Special Symbols Index

Real and Complex Analysis. By W. Rudin. Pp. 412. 84s. 1966. (McGraw-Hill, New York.)

This paper deals with optimal control of switched piecewise affine autonomous systems, where the objective is to minimize a performance index over an infinite time horizon. We assume that the switching sequence has a finite length, and that the decision variables are the switching instants and the sequence of operating modes. We present two different approaches for solving such an optimal control problem. The first approach iterates between a procedure that finds an optimal switching sequence of modes, and a procedure that finds the optimal switching instants. The second approach is inspired by dynamic programming and identifies the regions of the state space where an optimal mode switch should occur, therefore providing a state feedback control law.

Optimal control of continuous-time switched affine systems

This paper surveys recent results in the field of optimal control of hybrid and switched systems. We first summarize results that use different problem formulations and then explore the underlying relations among them. Specifically, based on the type of switching, we focus on two important classes of problems: internally forced switching (IFS) problems and externally forced switching (EFS) problems. For IFS problems, we focus on optimal control techniques for piecewise affine systems. For EFS problems, methodologies of two-stage optimization, embedding transformation and switching LQR design are investigated. Detailed optimization methods found in the literature are discussed.

Optimal control of hybrid switched systems: A brief survey

Necessary conditions were obtained by Shaikh and Caines (2002, 2003 and 2004) for hybrid optimal control problems (HOCPs) which resulted in a general hybrid maximum principle (HMP); further, a class of efficient, provably convergent hybrid maximum principle (HMP) algorithms were obtained based upon the HMP. The notion of optimality zones (OZs) (2004) was introduced as a theoretical framework enabling the computation of optimal schedules for HOCPs (i.e., discrete state sequences with the associated switching times and states). This paper presents the algorithm HMPZ which fully integrates the prior computation of the OZs into the HMP algorithms. Adding (i) the computational investment in the construction of the OZs for a given HOCP, and (ii) the complexity of the computation of the optimal schedule, optimal switching time and state sequence, and the optimal continuous control input, yields a complexity estimate for the algorithm (HMPZ) which is linear (i.e., O(L)) in the number of switching times L; this is to be compared with the geometric (i.e. O(|Q|L)) growth of a direct combinatoric search over the set of schedules, where Q denotes the discrete state set of the hybrid system

Optimality Zone Algorithms for Hybrid Systems Computation and Control: From Exponential to Linear Complexity

In [M.S. Shaikh, et al., 2002, April 2003, 2003] a class of hybrid optimal control problems was formulated and a set of necessary conditions for hybrid system trajectory optimally was presented. Employing these conditions, we presented and analyzed a class of general hybrid maximum principle (HMP) based algorithms for hybrid systems optimization. In this paper it is first shown how the HMP algorithm class can be extended with discrete search algorithms, which find locally optimal switching schedules and their associated, switching times. We then present the notion of optimality zones; these zones have a well defined geometrical structure and once they have been computed (or approximated) they permit the exponential complexity search for optimal schedule sequences of the first method to be reduced to a complexity level which under reasonable hypotheses is proportional to the number of zones. The algorithm HMP[Z] which performs this optimization is essentially a minor modification of the HMP algorithm and permits one to reach the global optimum in a single run of the HMP[MCS] algorithm. The efficacy of the proposed algorithms is illustrated via computational examples.

http://ieeexplore.ieee.org/iel5/8969/28482/01272935.pdf

On the optimal control of hybrid systems: analysis and zonal algorithms for trajectory and schedule optimization

In [1], [2], [3], [4] necessary conditions were obtained for hybrid optimal control problems (HOCPs) which resulted in a general Hybrid Maximum Principle (HMP); further, in [4], [5], a class of effficient, provably convergent Hybrid Maximum Principle (HMP) algorithms were obtained based up on the HMP. In [3], [4] the notion of optimality zones (OZs) was introduced as a theoretical framework for the computation of optimal location (i.e. discrete state) sequence for HOCPs (i.e. discrete state sequences with the associated switching times and states). This paper presents the algorithm HMPZ which fully integrates the prior computation of the OZs into the HMP algorithms of [4], [5]. Adding (a) the computational investment in the construction of the OZs for a given HOCP, and (b) the complexity of (i) the computation of the optimal schedule, (ii) the optimal switching time and optimal switching state sequence, and (iii) the optimal continuous control input, yields a complexity estimate for the algorithm (HMPZ) which is linear (i.e.O(L)) in the number of switching times L; this is to be compared with the geometric (i.e. O(|Q|L)) growth of a direct combinatoric search over the set of location sequence, where Q denotes the discrete state set of the hybrid system.

A class of the hybrid optimal control problem is formulated and a set of necessary conditions for hybrid system trajectory optimality is presented. These conditions constitute generalizations of the standard maximum principle (MP). Based upon these conditions, we propose a class of comparatively simple general hybrid maximum principle (HMP) algorithms for hybrid system optimization. Furthermore, the HMP algorithms are embedded in the so called HMP[Comb] algorithms class; the latter class extends the HMP class with discrete search algorithms which find locally optimal switching schedules and their associated switching times. The efficiency of the proposed algorithms is illustrated via a computational example.

M.S. Shaikh

Papers

On the Hybrid Optimal Control Problem: Theory and Algorithms

Optimality Zone Algorithms for Hybrid Systems Computation and Control: From Exponential to Linear Complexity

On the optimal control of hybrid systems: analysis and zonal algorithms for trajectory and schedule optimization

Optimality Zone Algorithms for Hybrid Systems Computation and Control: From Exponential to Linear Complexity

On the optimal control of hybrid systems: optimization of switching times and combinatoric location schedules