763 research outputs found
Discrete approximations for strict convex continuous time problems and duality
We propose a discrete approximation scheme to a class of Linear Quadratic Continuous Time Problems. It is shown, under positiveness of the matrix in the integral cost, that optimal solutions of the discrete problems provide a sequence of bounded variation functions which converges almost everywhere to the unique optimal solution. Furthermore, the method of discretization allows us to derive a number of interesting results based on finite dimensional optimization theory, namely, Karush-Kuhn-Tucker conditions of optimality and weak and strong duality. A number of examples are provided to illustrate the theory.8110
Higher Order Conditions in Nonlinear Optimal Control
The most widely used tool for the solution of optimal control problems is the Pontryagin
Maximum Principle. But the Maximum Principle is, in general, only a necessary
condition for optimality. It is therefore desirable to have supplementary conditions, for
example second order sufficient conditions, which confirm optimality (at least locally) of
an extremal arc, meaning one that satisfies the Maximum Principle.
Standard second order sufficient conditions for optimality, when they apply, yield the
information not only that the extremal is locally minimizing, but that it is also locally
unique. There are problems of interest, however, where minimizers are not locally unique,
owing to the fact that the cost is invariant under small perturbations of the extremal of
a particular structure (translations, rotations or time-shifting). For such problems the
standard second order conditions can never apply.
The first contribution of this thesis is to develop new second order conditions for
optimality of extremals which are applicable in some cases of interest when minimizers
are not locally unique. The new conditions can, for example, be applied to problems with
periodic boundary conditions when the cost is invariant under time translations.
The second order conditions investigated here apply to normal extremals. These
extremals satisfy the conditions of the Maximum Principle in normal form (with the
cost multiplier taken to be 1). It is, therefore, of interest to know when the Maximum
Principle applies in normal form. This issue is also addressed in this thesis, for optimal
control problems that can be expressed as calculus of variations problems. Normality of the
Maximum Principle follows from the fact that, under the regularity conditions developed,
the highest time derivative of an extremal arc is essentially bounded.
The thesis concludes with a brief account of possible future research directions
Analysis of Theoretical and Numerical Properties of Sequential Convex Programming for Continuous-Time Optimal Control
Sequential Convex Programming (SCP) has recently gained significant
popularity as an effective method for solving optimal control problems and has
been successfully applied in several different domains. However, the
theoretical analysis of SCP has received comparatively limited attention, and
it is often restricted to discrete-time formulations. In this paper, we present
a unifying theoretical analysis of a fairly general class of SCP procedures for
continuous-time optimal control problems. In addition to the derivation of
convergence guarantees in a continuous-time setting, our analysis reveals two
new numerical and practical insights. First, we show how one can more easily
account for manifold-type constraints, which are a defining feature of optimal
control of mechanical systems. Second, we show how our theoretical analysis can
be leveraged to accelerate SCP-based optimal control methods by infusing
techniques from indirect optimal control
Set-Valued Return Function and Generalized Solutions for Multiobjective Optimal Control Problems (MOC)
In this paper, we consider a multiobjective optimal control problem where the
preference relation in the objective space is defined in terms of a pointed
convex cone containing the origin, which defines generalized Pareto optimality.
For this problem, we introduce the set-valued return function V and provide a
unique characterization for V in terms of contingent derivative and
coderivative for set-valued maps, which extends two previously introduced
notions of generalized solution to the Hamilton-Jacobi equation for single
objective optimal control problems.Comment: 29 pages, submitted to SICO
A Nonsmooth Augmented Lagrangian Method and its Application to Poisson Denoising and Sparse Control
In this paper, fully nonsmooth optimization problems in Banach spaces with
finitely many inequality constraints, an equality constraint within a Hilbert
space framework, and an additional abstract constraint are considered. First,
we suggest a (safeguarded) augmented Lagrangian method for the numerical
solution of such problems and provide a derivative-free global convergence
theory which applies in situations where the appearing subproblems can be
solved to approximate global minimality. Exemplary, the latter is possible in a
fully convex setting. As we do not rely on any tool of generalized
differentiation, the results are obtained under minimal continuity assumptions
on the data functions. We then consider two prominent and difficult
applications from image denoising and sparse optimal control where these
findings can be applied in a beneficial way. These two applications are
discussed and investigated in some detail. Due to the different nature of the
two applications, their numerical solution by the (safeguarded) augmented
Lagrangian approach requires problem-tailored techniques to compute approximate
minima of the resulting subproblems. The corresponding methods are discussed,
and numerical results visualize our theoretical findings.Comment: 36 pages, 4 figures, 1 tabl
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