Philip D. Loewen

New Necessary Conditions for the Generalized Problem of Bolza

Problems of optimal control are considered in the neoclassical Bolza format, which centers on states and velocities and relies on nonsmooth analysis. Subgradient versions of the Euler--Lagrange equation and the Hamiltonian equation are shown to be necessary for the optimality of a trajectory, moreover in a newly sharpened form that makes these conditions equivalent to each other. At the same time, the assumptions on the Lagrangian integrand are weakened substantially over what has been required previously in obtaining such conditions.

Bolza Problems with General Time Constraints

This work provides necessary conditions for optimality in problems of optimal control expressed as instances of the generalized problem of Bolza, with the added feature that the fundamental planning interval is allowed to vary. A central product of the analysis is a generalization of the conservation-of-Hamiltonian condition for problems on either fixed or variable intervals. The results, which allow for unprecedented generality in the problem data, are derived from known properties of fixed-interval problems under the hypothesis that the time-dependence of the objective integrand has the same modest level of regularity as the state-dependence.

The adjoint arc in nonsmooth optimization

We extend the theory of necessary conditions for nonsmooth problems of Bolza in three ways: first, we incorporate state constraints of the intrinsic type x(t) E X(t) for all t; second, we make no assumption of calmness or normality; and third, we show that a single adjoint function of bounded variation simultaneously satisfies the Hamiltonian inclusion, the Euler-Lagrange inclusion, and the Weierstrass-Pontryagin maximum condition, along with the usual transversality relations.

The value function in optimal control: Sensitivity, controllability, and time-optimality

We consider a general optimal control problem in which the constraints depend on a parameter

Robust controller design for teleoperation systems

\alpha

The proximal normal formula in Hilbert space

, and the resulting value function

Differential inclusions with free time

V(\alpha )

Pontryagin-type necessary conditions for differential inclusion problems

. A formula for the generalized gradient of V is proven and then used to obtain results on stability and controllability of the problem. A special study is made of the time-optimal control problem, one consequence of which is a new criterion assuring local null-controllability of the system and continuity of the minimal time function at the origin.

State constraints in optimal control: A case study in proximal normal analysis

The controller design for a bilateral teleoperation system involves trade-offs between performance and robust stability. Beyond simple intuition, little is known how performance and robust stability trade off. This paper shows that it is possible to achieve robust stability and nominal performance of a bilateral teleoperation system by using a four-channel control architecture. The controller design problem is formulated as a multiple objective optimization problem, which is shown to be convex if parametrizing all stabilizing controllers via the Youla parametrization. Performance specifications, such as kinematic correspondence error, force tracking error, etc., are defined; and robust stability is also incorporated into the controller design. The controller design problem is formulated as a multiple objective optimization problem, which is shown to be convex if parametrizing all stabilizing controllers via the Youla parametrization. The limit of performance achievable with the designed controller, thus the exact form of the trade-offs between performance and robust stability can be computed numerically. To demonstrate those, this paper treats the design of a controller for a simple one degree-of-freedom (DOF) system model of a motion-scaling teleoperation system.

The range of the gradient of a Lipschitz C¹-smooth bump in infinite dimensions

On considere la formule normale proximale dans un espace de Hilbert. On presente une demonstration simple du theoreme de Borwein-Strojwas