Über differenzierbare asymptotisch konvexe funktionenfamilien bei der nicht-linearen gleichmäβigen approximation

Ohne Zusammenfassung Vorgelegt von L. Collatz     

On boundary controllability of a vibrating plate

A vibrating plate is here taken to satisfy the model equation:u _tt + ²u = 0 (where ²u:= (u); = Laplacian) with boundary conditions of the form:u _v = 0 and(u) _v = = control. Thus, the state is the pair [u, u _t] and controllability means existence of on := (0,T)× transfering any[u, u _t]₀ to any[u, u _t]_T. The formulation is given by eigenfunction expansion and duality. The substantive results apply to a rectangular plate. For largeT one has such controllability with = O(T ^–1/2). More surprising is that (based on a harmonic analysis estimate [11]) one has controllability for arbitrarily short times (in contrast to the wave equation:u _tt = u) with log = O(T ^–1) asT0. Some related results on minimum time control are also included.This research was partially supported under the grant AFOSR-82-0271.     

Optimal control of processes governed by partial differential equations part I: Heating processes

The primary concern of this paper is the treatment of optimal control of heating processes on the boundary of the spatial domains in which the processes take place. Given an initial state of the temperature at the timet=0 and a final state at some timeT>0, the problem is considered to find a control on the boundary by which the initial state is transferred to the final state. If in addition the control is subject to a restriction in absolute value, the question is answered under which conditions restricted control is possible in minimum time and time-minimal controls are characterized by a bang-bang-principle.As mathematical tool certain exponential moment problems are investigated and the results are applied.

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Zusammenfassung Diese Arbeit beschäftigt sich hauptsächlich mit optimaler Steuerung von Heizungsprozessen auf dem Rande der räumlichen Bereiche, in welchen die Prozesse stattfinden. Zu vorgegebenem Anfangszustand der Temperatur zur Zeitt=0 und vorgegebenem Endzustand zu einer ZeitT>0 wird das Problem betrachtet, eine Steuerung auf dem Rande zu finden, die den Anfangs- in den Endzustand überführt. Wird zusätzlich die Steuerung einer Beschränkung im Absolutwert unterworfen, so wird die Frage beantwortet, unter welchen Bedingungen beschränkte Steuerungen in minimaler Zeit existieren, und diese zeit-minimalen Steuerungen werden durch ein Bang-Bang-Prinzip charakterisiert.Als mathematisches Hilfsmittel werden gewisse Exponentialmomentenprobleme untersucht und die Ergebnisse werden angewandt.

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Invited survey.     

Numerical solution of minimum norm problems of distributed control of vibrations

This paper is concerned with the distributed control of a vibration process that can be described by a differential equation for a Hilbert-spacevalued functiony: [0, ) H. The control functions on the right-hand side of this equation are taken fromL ([0, ),H) equipped with the essential supremum norm. To be solved is the problem of time-minimal null-controllability by norm-bounded controls. This problem is essentially reduced to solving the problem of minimum norm control on a given time interval. This is solved via its dual problem which is approximately solved by truncation and discretization. Numerical results are presented for a vibrating string and a vibrating beam.