4,001 research outputs found
An indirect numerical method for a time-optimal state-constrained control problem in a steady two-dimensional fluid flow
This article concerns the problem of computing solutions to state-constrained
optimal control problems whose trajectory is affected by a flow field. This
general mathematical framework is particularly pertinent to the requirements
underlying the control of Autonomous Underwater Vehicles in realistic scenarii.
The key contribution consists in devising a computational indirect method which
becomes effective in the numerical computation of extremals to optimal control
problems with state constraints by using the maximum principle in Gamkrelidze's
form in which the measure Lagrange multiplier is ensured to be continuous. The
specific problem of time-optimal control of an Autonomous Underwater Vehicle in
a bounded space set, subject to the effect of a flow field and with bounded
actuation, is used to illustrate the proposed approach. The corresponding
numerical results are presented and discussed
Non-linear eigenvalue problems arising from growth maximization of positive linear dynamical systems
We study a growth maximization problem for a continuous time positive linear
system with switches. This is motivated by a problem of mathematical biology
(modeling growth-fragmentation processes and the PMCA protocol). We show that
the growth rate is determined by the non-linear eigenvalue of a max-plus
analogue of the Ruelle-Perron-Frobenius operator, or equivalently, by the
ergodic constant of a Hamilton-Jacobi (HJ) partial differential equation, the
solutions or subsolutions of which yield Barabanov and extremal norms,
respectively. We exploit contraction properties of order preserving flows, with
respect to Hilbert's projective metric, to show that the non-linear eigenvector
of the operator, or the "weak KAM" solution of the HJ equation, does exist. Low
dimensional examples are presented, showing that the optimal control can lead
to a limit cycle.Comment: 8 page
The turnpike property in finite-dimensional nonlinear optimal control
Turnpike properties have been established long time ago in finite-dimensional
optimal control problems arising in econometry. They refer to the fact that,
under quite general assumptions, the optimal solutions of a given optimal
control problem settled in large time consist approximately of three pieces,
the first and the last of which being transient short-time arcs, and the middle
piece being a long-time arc staying exponentially close to the optimal
steady-state solution of an associated static optimal control problem. We
provide in this paper a general version of a turnpike theorem, valuable for
nonlinear dynamics without any specific assumption, and for very general
terminal conditions. Not only the optimal trajectory is shown to remain
exponentially close to a steady-state, but also the corresponding adjoint
vector of the Pontryagin maximum principle. The exponential closedness is
quantified with the use of appropriate normal forms of Riccati equations. We
show then how the property on the adjoint vector can be adequately used in
order to initialize successfully a numerical direct method, or a shooting
method. In particular, we provide an appropriate variant of the usual shooting
method in which we initialize the adjoint vector, not at the initial time, but
at the middle of the trajectory
Discrete Approximations of a Controlled Sweeping Process
The paper is devoted to the study of a new class of optimal control problems
governed by the classical Moreau sweeping process with the new feature that the polyhe-
dral moving set is not fixed while controlled by time-dependent functions. The dynamics of
such problems is described by dissipative non-Lipschitzian differential inclusions with state
constraints of equality and inequality types. It makes challenging and difficult their anal-
ysis and optimization. In this paper we establish some existence results for the sweeping
process under consideration and develop the method of discrete approximations that allows
us to strongly approximate, in the W^{1,2} topology, optimal solutions of the continuous-type
sweeping process by their discrete counterparts
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