1,721,004 research outputs found
The extended Euler-Lagrange condition for nonconvex variational problems
No full textThis paper provides necessary conditions of optimality for a general variational problem for which the dynamic constraint is a differential inclusion with a possibly nonconvex right side. They take the form of an Euler-Lagrange inclusion involving convexification in only one coordinate, supplemented by the transversality and Weierstrass conditions. It is also shown that for time-invariant, free time problems, the adjoint arc can be chosen so that the Hamiltonian function is constant along the minimizing state arc. The methods used here, based on simple "finite dimensional" nonsmooth calculus, Clarke decoupling, and a rudimentary version of the maximum principle, offer an alternative, and somewhat simpler, derivation of such results to those used by Ioffe and Rockafellar in concurrent research
Degenerate optimal control problems with state constraints
No full textStandard necessary conditions for optimal control problems with pathwise state constraints supply no useful information about minimizers in a number of cases of interest, e.g., when the left endpoint of state trajectories is fixed at x(0) and x(0) lies in the boundary of the state constraint set; in these cases a nonzero, but nevertheless trivial, set of multipliers exists. We give conditions for the existence of nontrivial multipliers. A feature of these conditions is that they allow nonconvex velocity sets and measurably time-dependent data. The proof techniques are based on refined estimates of the distance of a given state trajectory from the set of state trajectories satisfying the state constraint, originating in the dynamic programming literature
A theorem on existence of neighbouring trajectories satisfying a state constraint, with applications to optimal control.
No full textThis paper deals with the implementation of a predictor control law and the suboptimal digital redesign for a continuous-time plant with a delayed input. It is shown that the closed-loop eigenvalues of the delayed system designed via a predicted-state feedback controller are identical to those of the non-delayed system with the same gain matrix based on the non-predicted state feedback. Thus, the well-developed design methods of linear quadratic regulators for continuous-time non-delayed systems can be indirectly applied to delayed-input systems. Moreover, for practical implementation by digital computer, we derive a suboptimal digital redesign of the delayed-input system using the predictor control
Discontinuous solutions to the Hamilton Jacobi equation under second order interiority hypotheses
Get PDFThis paper concerns the characterization of a lower semicontinuous value function in terms of the unique generalized solution of the Hamilton Jacobi equation (HJE), for optimal control problems with end point and pathwise state constraints. There is an extensive literature on this topic, in the contexts both of finite and infinite horizon problems. A key step in establishing this link is to validate a certain 'interiority' property, specifically to show that an arbitrary state trajectory satisfying the state constraint can be approximated by an interior state trajectory. The step has traditionally been accomplished under the hypothesis that, at each point in the boundary of the state constraint set, there exist admissible velocities strictly pointing away from the boundary of the state constraint set. This is a significant restriction, since this 'strict sense' outward/inward pointing condition is violated by control of systems, typically encountered in mechanical control, where the control action affects the rate of change of the state, not directly, but through the intermediary of a dynamic system. A class of control systems has been recently identified that enjoy this interiority property, but for which the strict sense outward pointing condition is violated. These advances have been used to establish uniqueness of continuous viscosity solutions of the relevant Hamilton Jacobi equation for an infinite horizon formulation of the problem involving such systems. In this paper, we exploit this newly discovered property also to provide the desired characterization of the value function for finite horizon problems involving endpoint and pathwise state constraints, when the value function is possibly discontinuous and the traditional strict outward pointing condition is replaced by a softer, second order condition
Optimal impulse control problems with time delays: An illustrative example
Get PDFFor impulse control systems described by a measure driven differential equation, depending linearly on the measure, it is customary to interpret the state trajectory corresponding to an impulse control, specified by a measure, as the limit of state trajectories associated with some sequence of conventional controls approximating the measure. It is known that, when the measure is vector valued, it is possible that different choices of approximating sequences for the measure give rise to different limiting state trajectories. If the measure is scalar valued, however, there is a unique limiting trajectory. Now consider impulse control systems, in which the right side of the measure driven differential equation depends on both the current and delayed states. In recent work by the authors it has been shown that, for such impulse control systems with time delay, the state trajectory corresponding to a given measure may be non-unique, even when the measure is scalar valued. It was also shown that each limiting state trajectory can be identified with the unique state trajectory associated with some measure together with a family of ‘attached controls’. (The attached controls capture the nature of the measure approximation.) The authors also derived a maximum principle governing minimizers for a general class of impulse optimal control problems with time delay, in which the domain of the optimization problem comprises measures coupled with a family of ‘attached controls’. The purpose of this paper is both to illustrate, by means of an example, this newly discovered non-uniqueness phenomenon and to provide the first application of the new maximum principle, to investigate minimizers for scalar input impulse optimal control problems with time delay, in circumstances when limiting state trajectories associated with a given measure control are not unique. The example is an optimal control problem, for which the underlying control system is a forced harmonic oscillator, with scalar impulse control, in which the control gain is a nonlinear function of the current and delayed states
OPTIMAL IMPULSIVE CONTROL FOR TIME DELAY SYSTEMS
Get PDFWe introduce discontinuous solutions to nonlinear impulsive control systems with state time delays in the dynamics and derive necessary optimality conditions in the form of a max- imum principle for associated optimal control problems. In the case without delays, if the measure control is scalar valued, the corresponding discontinuous state trajectory, understood as a limit of classical state trajectories for absolutely continuous controls approximating the measure, is unique. For vector-valued measure controls, however, the limiting trajectory is not unique and a full de- scription of the control must include additional ``attached"" controls affecting instantaneous state evolution at a discontinuity. For impulsive control systems with time delays we reveal a new phe- nomenon, namely, that the limiting state trajectory resulting from different approximations of a given measure control needs not to be unique, even in the scalar case. Correspondingly, our framework allows for additional attached controls, even though the measure control is scalar valued
Normality and Gap Phenomena in Optimal Unbounded Control
Get PDFOptimal unbounded control problems with affine control dependence may fail to
have minimizers in the class of absolutely continuous state trajectories. For
this reason, extended impulsive versions --which cannot be of
measure-theoretical type-- have been investigated, in which the domain is
enlarged to include discontinuous state trajectories of bounded variation, and
for which existence of minimizers is guaranteed. It is of interest to know
whether the passage from the original optimal control problem to its extension
introduces an infimum gap. This paper provides sufficient conditions for the
absence of an infimum gap based on normality of extremals. In certain cases,
the normality conditions reduce to simple verifiable criteria, which improve on
earlier, directly-derived sufficient conditions for no infimum gap
Decomposition of Differential Games with Multiple Targets
Get PDFThis paper provides a decomposition technique for the purpose of simplifying the solution of certain zero-sum differential games. The games considered terminate when the state reaches a target, which can be expressed as the union of a collection of target subsets considered as ‘multiple targets’; the decomposition consists in replacing the original target by each of the target subsets. The value of the original game is then obtained as the lower envelope of the values of the collection of games, resulting from the decomposition, which can be much easier to solve than the original game. Criteria are given for the validity of the decomposition. The paper includes examples, illustrating the application of the technique to pursuit/evasion games and to flow control
Going Beyond Counting First Authors in Author Co-citation Analysis
Get PDFThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
- …
