The Derivation of Cubic Splines with Obstacles by Methods of Optimization and Optimal Control.
The Euler equation for a class of nonconvex problems.
The Lazy Travelling Salesman Problem in
We study a parameter (σ) dependent relaxation of the Travelling Salesman Problem on . The relaxed problem is reduced to the Travelling Salesman Problem as 0. For increasing σ it is also an ordered clustering algorithm for a set of points in . A dual formulation is introduced, which reduces the problem to a convex optimization, provided the minimizer is in the domain of convexity of the relaxed functional. It is shown that this last condition is generically satisfied, provided σ is large enough. ...
The Monge problem for strictly convex norms in
We prove the existence of an optimal transport map for the Monge problem in a convex bounded subset of under the assumptions that the first marginal is absolutely continuous with respect to the Lebesgue measure and that the cost is given by a strictly convex norm. We propose a new approach which does not use disintegration of measures.
The problem of the body of revolution of minimal resistance
Newton's problem of the body of minimal aerodynamic resistance is traditionally stated in the class of convex axially symmetric bodies with fixed length and width. We state and solve the minimal resistance problem in the wider class of axially symmetric but generally nonconvex bodies. The infimum in this problem is not attained. We construct a sequence of bodies minimizing the resistance. This sequence approximates a convex body with smooth front surface, while the surface of approximating bodies...
The shooting approach in analyzing bang-bang extremals with simultaneous control switches
The squares of the Laplacian-Dirichlet eigenfunctions are generically linearly independent
The paper deals with the genericity of domain-dependent spectral properties of the Laplacian-Dirichlet operator. In particular we prove that, generically, the squares of the eigenfunctions form a free family. We also show that the spectrum is generically non-resonant. The results are obtained by applying global perturbations of the domains and exploiting analytic perturbation properties. The work is motivated by two applications: an existence result for the problem of maximizing the rate of...
Time-optimal control of infinite order hyperbolic systems with time delays
In this paper, the time-optimal control problem for infinite order hyperbolic systems in which time delays appear in the integral form both in state equations and in boundary conditions is considered. Optimal controls are characterized in terms of an adjoint system and shown to be unique and bang-bang. These results extend to certain cases of nonlinear control problems. The particular properties of optimal control are discussed.