Geodesic
Optimization of the Boundary Control of an Elastic Force at One Endpoint of a String with the Other Endpoint Fixed
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Studies on function theory and differential equations, Tome 248 (2005), pp. 117-123 
V. A. Il'in. Optimization of the Boundary Control of an Elastic Force at One Endpoint of a String with the Other Endpoint Fixed. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Studies on function theory and differential equations, Tome 248 (2005), pp. 117-123. http://geodesic.mathdoc.fr/item/TM_2005_248_a11/
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     author = {V. A. Il'in},
     title = {Optimization of the {Boundary} {Control} of an {Elastic} {Force} at {One} {Endpoint} of a {String} with the {Other} {Endpoint} {Fixed}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
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Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

For a large time interval $T$, we study the boundary control of an elastic force at the endpoint $x=0$ of a string under the condition that the second endpoint $x=l$ of the string is fixed. We determine and present in an analytical form the optimal boundary control $u_x(0,t)=\mu (t)$ that minimizes the elastic boundary energy integral $\int _0^T\mu ^2(t)\,dt$ over the set of all functions $\mu (t)$ from the class $L_2[0,T]$ under the condition that the oscillation process transfers the string from an arbitrary given initial state into an arbitrary given final state.

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