Controllability,extremality, and abnormality in nonsmooth optimal control

We derive sufficient conditions for controllability and necessary conditions for minimum in nonsmooth optimal control problems defined by differential or functional-integral equations with isoperimetric and unilateral restrictions. We consider the cases when the controls are relaxed or chosen fromabundant sets of original (ordinary) controls (which include most, or all, of the control sets studied in the literature). We prove that, if there exist optimal strictly original controls (that is, controls that are optimal in an abundant set but not among relaxed controls), then the problem admits abnormal extremals. We also study the abnormality of the optimal strictly original controls themselves.     

Local Minimum Principle for Optimal Control Problems Subject to Differential-Algebraic Equations of Index Two

Necessary conditions in terms of a local minimum principle are derived for optimal control problems subject to index-2 differential-algebraic equations, pure state constraints, and mixed control-state constraints. Differential-algebraic equations are composite systems of differential equations and algebraic equations, which arise frequently in practical applications. The local minimum principle is based on the necessary optimality conditions for general infinite optimization problems. The special structure of the optimal control problem under consideration is exploited and allows us to obtain more regular representations for the multipliers involved. An additional Mangasarian-Fromowitz-like constraint qualification for the optimal control problem ensures the regularity of a local minimum. An illustrative example completes the article.The author thanks the referees for careful reading and helpful suggestions and comments.     

Numerical Computation of Optimal Trajectories for Coplanar, Aeroassisted Orbital Transfer

This paper is concerned with the problem of the optimal coplanaraeroassisted orbital transfer of a spacecraft from a high Earth orbitto a low Earth orbit. It is assumed that the initial and final orbits arecircular and that the gravitational field is central and is governed by theinverse square law. The whole trajectory is assumed to consist of twoimpulsive velocity changes at the begin and end of one interior atmosphericsubarc, where the vehicle is controlled via the lift coefficient.The problem is reduced to the atmospheric part of the trajectory, thusarriving at an optimal control problem with free final time and liftcoefficient as the only (bounded) control variable. For this problem,the necessary conditions of optimal control theory are derived. Applyingmultiple shooting techniques, two trajectories with different controlstructures are computed. The first trajectory is characterized by a liftcoefficient at its minimum value during the whole atmospheric pass. For thesecond trajectory, an optimal control history with a boundary subarcfollowed by a free subarc is chosen. It turns out, that this secondtrajectory satisfies the minimum principle, whereas the first one fails tosatisfy this necessary condition; nevertheless, the characteristicvelocities of the two trajectories differ only in the sixth significantdigit.In the second part of the paper, the assumption of impulsive velocitychanges is dropped. Instead, a more realistic modeling with twofinite-thrust subarcs in the nonatmospheric part of the trajectory isconsidered. The resulting optimal control problem now describes the wholemaneuver including the nonatmospheric parts. It contains as controlvariables the thrust, thrust angle, and lift coefficient. Further,the mass of the vehicle is treated as an additional state variable. For thisoptimal control problem, numerical solutions are presented. They are comparedwith the solutions of the impulsive model.     

A globally convergent semi-smooth Newton method for control-state constrained DAE optimal control problems

We investigate a semi-smooth Newton method for the numerical solution of optimal control problems subject to differential-algebraic equations (DAEs) and mixed control-state constraints. The necessary conditions are stated in terms of a local minimum principle. By use of the Fischer-Burmeister function the local minimum principle is transformed into an equivalent nonlinear and semi-smooth equation in appropriate Banach spaces. This nonlinear and semi-smooth equation is solved by a semi-smooth Newton method. We extend known local and global convergence results for ODE optimal control problems to the DAE optimal control problems under consideration. Special emphasis is laid on the calculation of Newton steps which are given by a linear DAE boundary value problem. Regularity conditions which ensure the existence of solutions are provided. A regularization strategy for inconsistent boundary value problems is suggested. Numerical illustrations for the optimal control of a pendulum and for the optimal control of discretized Navier-Stokes equations conclude the article.     

Optimal control of constrained time lag systems: Necessary conditions and numerical treatment

We consider retarded optimal control problems with constant delays in state and control variables under mixed controlstate inequality constraints. First order necessary optimality conditions in the form of Pontryagin's minimum principle are presented and discussed as well as numerical methods based upon discretization techniques and nonlinear programming. The minimum principle for the considered problem class leads to a boundary value problem which is retarded in the state dynamics and advanced in the costate dynamics. It can be shown that the Lagrange multipliers associated with the programming problem provide a consistent discretization of the advanced adjoint equation for the delayed control problem. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Algorithms for Unconstrained Optimization Problems via Control Theory

Existing algorithms for solving unconstrained optimization problems are generally only optimal in the short term. It is desirable to have algorithms which are long-term optimal. To achieve this, the problem of computing the minimum point of an unconstrained function is formulated as a sequence of optimal control problems. Some qualitative results are obtained from the optimal control analysis. These qualitative results are then used to construct a theoretical iterative method and a new continuous-time method for computing the minimum point of a nonlinear unconstrained function. New iterative algorithms which approximate the theoretical iterative method and the proposed continuous-time method are then established. For convergence analysis, it is useful to note that the numerical solution of an unconstrained optimization problem is none other than an inverse Lyapunov function problem. Convergence conditions for the proposed continuous-time method and iterative algorithms are established by using the Lyapunov function theorem.     

TIME OPTIMAL IMPLUSE CONTROL PROBLEMS

ＴＩＭＥ　ＯＰＴＩＭＡＬ　ＩＭＰＬＵＳＥ　ＣＯＮＴＲＯＬ　ＰＲＯＢＬＥＭＳＴＩＭＥＯＰＴＩＭＡＬＩＭＰＬＵＳＥＣＯＮＴＲＯＬＰＲＯＢＬＥＭＳ￥ＸｕＺｏｎｇｙｕｎ（ＦｕｄａｎＵｎｉｖｅｒｓｉｔｙ，）Ａｂｓｔｒａｃｔ：Ｉｎｔｈｉｓｐａｐｅｒ，ａｃｌａｓｓ...     

Application of the hypodifferential descent method to the problem of constructing an optimal control

In this paper the problem of optimal control of a nonlinear ODE system with given boundary conditions and the integral restriction on control is considered. With the help of the theory of exact penalty functions the original problem is reduced to the problem of unconstrained minimization of a nonsmooth functional. The necessary minimum conditions in terms of hypodifferentials are found. A class of problems for which these conditions are also sufficient is distinguished. On the basis of these conditions the hypodifferential descent method is applied to the considered problem. Under some additional assumptions the hypodifferential descent method converges in a certain sense.     

Sufficient quadratic conditions of extremum for discontinuous controls in optimal control problems with mixed constraints

We derive second-order sufficient optimality conditions for discontinuous controls in optimal control problems of ordinary differential equations with initial-final state constraints and mixed state-control constraints of equality and inequality type. Under the assumption that the gradients with respect to the control of active mixed constraints are linearly independent, the sufficient conditions imply a bounded strong minimum in the problem.     

Convergence Analysis of the Implicit Euler-discretization and Sufficient Conditions for Optimal Control Problems Subject to Index-one Differential-algebraic Equations

For optimal control problems subject to index-one differential-algebraic equations in semi-explicit form we discuss second order sufficient conditions in form of a coercivity condition taking into account the two-norm discrepancy. Furthermore we introduce a related Riccati-type and Legendre-Clebsch condition which are sufficient for the validity of the coercivity condition. Using the implicit Euler-discretization we approximate the optimal control problem and analyze the convergence of solutions of the local minimum principle for the discretized optimal control problem by applying the general convergence framework of Stetter, which requires the discretization method to be continuous, consistent, and stable.

     

An optimal control problem with a random stopping time

This paper deals with a stochastic optimal control problem where the randomness is essentially concentrated in the stopping time terminating the process. If the stopping time is characterized by an intensity depending on the state and control variables, one can reformulate the problem equivalently as an infinite-horizon optimal control problem. Applying dynamic programming and minimum principle techniques to this associated deterministic control problem yields specific optimality conditions for the original stochastic control problem. It is also possible to characterize extremal steady states. The model is illustrated by an example related to the economics of technological innovation.This research has been supported by NSERC-Canada, Grants 36444 and A4952; by FCAR-Québec, Grant 88EQ3528, Actions Structurantes; and by MESS-Québec, Grant 6.1/7.4(28).     

Optimal birth control of population dynamics. II. Problems with free final time,phase constraints,and mini-max costs

A previous analysis of optimal birth control of population systems of the McKendrick type (a distributed parameter system involving 1st order partial differential equations with nonlocal bilinear boundary control) raised 3 additional issues--free final time problem, system with phase constraints, and the mini-max control problem of a population. The free final time problem considers the minimum time problem to be a special case, but relaxes many convexity assumptions. Theorems (maximum principles) and corollaries are developed that flow from the terminology and mathematical notations set forth in the earlier article.     

New general guidance method in constrained optimal control,part 1: Numerical method

A very fast numerical method is developed for the computation of neighboring optimum feedback controls. This method is applicable to a general class of optimal control problems (for example, problems including inequality constraints and discontinuities) and needs no on-line computation, except for one matrix-vector multiplication. The method is based on the so-called accessory minimum problem. The necessary conditions for this auxiliary optimal control problem form a linear multipoint boundary-value problem with linear jump conditions, which is especially well suited for numerical treatment. In the second part of this paper, the performance of the guidance scheme is shown for the heating-constrained cross-range maximization problem of a space-shuttle-orbiter-type vehicle.This research was supported in part by the Deutsche Forschungsgemeinschaft under the Schwerpunktprogramm Anwendungsbezogene Optimierung und Steuerung.The authors wish to express their sincere and grateful appreciation to Professor Roland Bulirsch who encouraged this work.     

Optimality Conditions and Synthesis for the Minimum Time Problem

We study the minimum time optimal control problem for a nonlinear system in R ⁿ with a general target. Necessary and sufficient optimality conditions are obtained. In particular, we describe a class of costates that are included in the superdifferential of the minimum time function, even in the case when this function is only lower semicontinuous. Two set-valued maps are constructed to provide time optimal synthesis.     

Sufficient conditions for a strong relative minimum in an optimal control problem

In this paper, on the basis of Young's method (Ref. 1), sufficient conditions for a strong relative minimum in an optimal control problem are given. Young's method generalizes geodesic coverings and the simplest Hilbert integral from the standard variational calculus. This paper carries Young's method over to nonparametric problems.     

Application of relaxed solutions to minimum sensitivity optimal control

Relaxed variational techniques are applied to a minimum sensitivity control problem. Sensitivity of a trajectory is minimized to perturbations in initial conditions. Rather than using the optimal control that does indeed exist and that satisfies the final conditions exactly, a suboptimal control is used that transfers the system from the given initial state to an arbitrary small neighborhood of the given final state, and that results in a considerably better performance than the optimal solution. The suboptimal control is constructed using the optimal controls of the relaxed problem.This paper is based upon the Ph.D. dissertation by the author at Purdue University, Lafayette, Indiana. The author wishes to thank Professor Violet B. Haas, School of Electrical Engineering, Purdue University, for introducing him to relaxed variational problems and for many very helpful suggestions through the course of this work.     

An optimal policy for a fish harvest

A dynamical model for harvesting a fish population system is proposed by introducing control into the known Verhulst-Pearl model. An optimal control problem including some parameters is stated, and the usual necessary conditions are applied. For specific parameter values, the candidate control policy is deduced, and optimality is verified by applying a sufficiency theorem. The optimal trajectories may contain maximum and minimum control arcs as well as a singular subarc. The significance of the singular arc is interpreted in terms of the system dynamics.     

Instability of optimal equilibria in the minimum mass design of uniform shallow arches

Necessary and sufficient conditions for the minimum mass design of arbitrarily loaded uniform shallow arches are derived. The problem is posed as an optimal control problem with mass as the criterion, initial curvature and axial load as design variables, and with the differential equations of axial and transverse equilibrium of the arch as side conditions. Thus, an optimal equilibrium is associated with each optimal design, and the stability of these equilibria becomes an integral part of the problem solution. As an example, the design process is carried out for the sinusoidally loaded hinged-hinged arch with a fixed span. It turns out that, depending on the given load amplitude, the optimal equilibrium can be unstable, stable after snap-through, and nonunique with one equilibrium unstable and the other stable after snap-through, at the design load of the arch. In addition, a necessary condition for a local minimum is the same as the usual critical point condition in stability analysis, thus assuring the instability of the arch at the optimum. A brief survey of earlier work on the optimal design of arches and curved beams is also included.     

Piecewise-Linear Pathways to the Optimal Solution Set in Linear Programming

An optimal control problem with four linear controls describing a sophisticated concern model is investigated. The numerical solution of this problem by combination of a direct collocation and an indirect multiple shooting method is presented and discussed. The approximation provided by the direct method is used to estimate the switching structure caused by the four controls occurring linearly. The optimal controls have bang-bang subarcs as well as constrained and singular subarcs. The derivation of necessary conditions from optimal control theory is aimed at the subsequent application of an indirect multiple shooting method but is also interesting from a mathematical point of view. Due to the linear occurrence of the controls, the minimum principle leads to a linear programming problem. Therefore, the Karush–Kuhn–Tucker conditions can be used for an optimality check of the solution obtained by the indirect method.     

Abort landing in windshear: Optimal control problem with third-order state constraint and varied switching structure

Optimal abort landing trajectories of an aircraft under different windshear-downburst situations are computed and discussed. In order to avoid an airplane crash due to severe winds encountered by the aircraft during the landing approach, the minimum altitude obtained during the abort landing maneuver is to be maximized. This maneuver is mathematically described by a Chebyshev optimal control problem. By a transformation to an optimal control problem of Mayer type, an additional state variable inequality constraint for the altitude has to be taken into account; here, its order is three. Due to this altitude constraint, the optimal trajectories exhibit, depending on the windshear parameters, up to four touch points and also up to one boundary arc at the minimum altitude level. The control variable is the angle of attack time rate which enters the equations of motion linearly; therefore, the Hamiltonian of the problem is nonregular. The switching structures also includes up to three singular subarcs and up to two boundary subarcs of an angle of attack constraint of first order. This structure can be obtained by applying some advanced necessary conditions of optimal control theory in combination with the multiple-shooting method. The optimal solutions exhibit an oscillatory behavior, reaching the minimum altitude level several times. By the optimization, the maximum survival capability can also be determined; this is the maximum wind velocity difference for which recovery from windshear is just possible. The computed optimal trajectories may serve as benchmark trajectories, both for guidance laws that are desirable to approach in actual flight and for optimal trajectories may then serve as benchmark trajectories both for guidance schemes and also for numerical methods for problems of optimal control.This paper is dedicated to Professor George Leitmann on the occasion of his seventieth birthday.