250 research outputs found
Strict dissipativity for discrete time discounted optimal control problems
The paradigm of discounting future costs is a common feature of economic applications of optimal control. In this paper, we provide several results for such discounted optimal control aimed at replicating the now wellknown results in the standard, undiscounted, setting whereby (strict) dissipa-tivity, turnpike properties, and near-optimality of closed-loop systems using model predictive control are essentially equivalent. To that end, we introduce a notion of discounted strict dissipativity and show that this implies various properties including the existence of available storage functions, required sup-ply functions, and robustness of optimal equilibria. Additionally, for discount factors sufficiently close to one we demonstrate that strict dissipativity implies discounted strict dissipativity and that optimally controlled systems, derived from a discounted cost function, yield practically asymptotically stable equi-libria. Several examples are provided throughout
On a discounted notion of strict dissipativity
Recent results in the literature have provided connections between the so-called turnpike property, near optimality of closed-loop solutions using model predictive control schemes, and strict dissipativity. An important feature of these results is that strict dissipativity provides a checkable condition for the other two properties. These results relate to optimal control problems with undiscounted stage cost. Motivated by applications in economics, we consider optimal control problems with discounted stage cost and define a notion of discounted strict dissipativity. As in the undiscounted case, we show that discounted strict dissipativity provides a checkable condition for various properties of the solutions of the optimal control problem associated with the appropriately defined discounted available storage function
Local turnpike analysis using local dissipativity for discrete time discounted optimal control
Recent results in the literature have provided connections between the
so-called turnpike property, near optimality of closed-loop solutions, and
strict dissipativity. Motivated by applications in economics, optimal control
problems with discounted stage cost are of great interest. In contrast to
non-discounted optimal control problems, it is more likely that several
asymptotically stable optimal equilibria coexist. Due to the discounting and
transition cost from a local to the global equilibrium, it may be more
favourable staying in a local equilibrium than moving to the global - cheaper -
equilibrium. In the literature, strict dissipativity was shown to provide
criteria for global asymptotic stability of optimal equilibria and turnpike
behavior. In this paper, we propose a local notion of discounted strict
dissipativity and a local turnpike property, both depending on the discount
factor. Using these concepts, we investigate the local behaviour of
(near-)optimal trajectories and develop conditions on the discount factor to
ensure convergence to a local asymptotically stable optimal equilibrium
On the Stabilization and Stability Robustness Against Small Delays of Some Damped wave equations
Cataloged from PDF version of article.In this note we consider a system which can be modeled
by two different one-dimensional damped wave equations in a bounded
domain, both parameterized by a nonnegative damping constant. We
assume that the system is fixed at one end and is controlled by a boundary
controller at the other end. We consider two problems, namely the
stabilization and the stability robustness of the closed-loop system against
arbitrary small time delays in the feedback loop. We propose a class
of dynamic boundary controllers and show that these controllem solve
the stabilization problem when the damping cuefMent is nonnegative
and stability robustness problem when the damping coefficient is strictly
positive
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