scholarly journals Existence theorems in multidimensional problems of optimization with distributed and boundary controls

1972 ◽  
Vol 46 (5) ◽  
pp. 321-355 ◽  
Author(s):  
Lamberto Cesari ◽  
David E. Cowles
Keyword(s):  
Existence Theorems ◽  
Multidimensional Problems ◽  
Boundary Controls ◽  
Distributed And Boundary Controls

scholarly journals Generalized control with compact support for systems with distributed parameters

2015 ◽  
Vol 25 (1) ◽  
pp. 5-20 ◽  
Author(s):  
Asatur ZH. Khurshudyan
Keyword(s):  
Compact Support ◽  
Exact Controllability ◽  
Time Interval ◽  
State Function ◽  
Infinite Dimensional ◽  
Distributed Parameters ◽  
Boundary Controls ◽  
Systems With Distributed Parameters ◽  
Distributed And Boundary Controls ◽  
Controllability Conditions

Abstract We propose a generalization of the Butkovskiy's method of control with compact support [1] allowing to derive exact controllability conditions and construct explicit solutions in control problems for systems with distributed parameters. The idea is the introduction of a new state function which is supported in considered bounded time interval and coincides with the original one therein. By means of techniques of the distributions theory the problem is reduced to an interpolation problem for Fourier image of unknown function or to corresponding system of integral equalities. Treating it as infinite dimensional problem of moments, its L1, L2 and L∞-optimal solutions are constructed explicitly. The technique is explained for semilinear wave equation with distributed and boundary controls. Particular cases are discussed.

Hierarchical exact controllability of semilinear parabolic equations with distributed and boundary controls

2019 ◽  
Vol 22 (07) ◽  
pp. 1950034 ◽  
Author(s):  
F. D. Araruna ◽  
E. Fernández-Cara ◽  
L. C. da Silva
Keyword(s):  
Parabolic Equations ◽  
Exact Controllability ◽  
Hierarchic Control ◽  
Semilinear Problems ◽  
Nash Strategies ◽  
Boundary Controls ◽  
Equilibrium Pair ◽  
Small Parts ◽  
Semilinear Parabolic ◽  
Distributed And Boundary Controls

We present some exact controllability results for parabolic equations in the context of hierarchic control using Stackelberg–Nash strategies. We analyze two cases: in the first one, the main control (the leader) acts in the interior of the domain and the secondary controls (the followers) act on small parts of the boundary; in the second one, we consider a leader acting on the boundary while the followers are of the distributed kind. In both cases, for each leader, an associated Nash equilibrium pair is found; then, we obtain a leader that leads the system exactly to a prescribed (but arbitrary) trajectory. We consider linear and semilinear problems.

Teichmuller Geometry

2017 ◽  
Author(s):  
Benson Farb ◽  
Dan Margalit
Keyword(s):  
Uniqueness Theorem ◽  
Existence Theorems ◽  
Quadratic Differentials ◽  
Complex Structures ◽  
Metric Geometry ◽  
Quasiconformal Maps ◽  
Hyperbolic Structures ◽  
Teichmüller Metric ◽  
Uniqueness And Existence ◽  
Teichmüller Mapping

This chapter focuses on the metric geometry of Teichmüller space. It first explains how one can think of Teich(Sɡ) as the space of complex structures on Sɡ. To this end, the chapter defines quasiconformal maps between surfaces and presents a solution to the resulting Teichmüller's extremal problem. It also considers the correspondence between complex structures and hyperbolic structures, along with the Teichmüller mapping, Teichmüller metric, and the proof of Teichmüller's uniqueness and existence theorems. The fundamental connection between Teichmüller's theorems, holomorphic quadratic differentials, and measured foliations is discussed as well. Finally, the chapter describes the Grötzsch's problem, whose solution is tied to the proof of Teichmüller's uniqueness theorem.

Interpolation and extrapolation

2018 ◽  
Author(s):  
Joseph F. Boudreau ◽  
Eric S. Swanson
Keyword(s):  
Public Transportation ◽  
Spline Interpolation ◽  
Lattice Gauge Theory ◽  
Human Vision ◽  
Physical Sciences ◽  
One Dimensional ◽  
Lattice Gauge ◽  
Multidimensional Problems ◽  
Data Points ◽  
Lagrange Interpolating Polynomial

This chapter deals with two related problems occurring frequently in the physical sciences: first, the problem of estimating the value of a function from a limited number of data points; and second, the problem of calculating its value from a series approximation. Numerical methods for interpolating and extrapolating data are presented. The famous Lagrange interpolating polynomial is introduced and applied to one-dimensional and multidimensional problems. Cubic spline interpolation is introduced and an implementation in terms of Eigen classes is given. Several techniques for improving the convergence of Taylor series are discussed, including Shank’s transformation, Richardson extrapolation, and the use of Padé approximants. Conversion between representations with the quotient-difference algorithm is discussed. The exercises explore public transportation, human vision, the wine market, and SU(2) lattice gauge theory, among other topics.

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