ACTA issues

Classical solutions to observation problems for infinite strings under minimally smooth force

András Szijártó, Jenő Hegedűs

Acta Sci. Math. (Szeged) 81:3-4(2015), 503-526

Abstract. We study the transversal vibrations $u=u(x,t)\in C^2(\mathbb{R}^2)$ of the infinite $x\in(-\infty,\infty )$ string under the external force $f(x,t)$ for all $t \in(-\infty,\infty )$, when the classical D'Alembert's formula with Duhamel's principle describes the whole vibration process using the initial data $u|_{t=0}=\varphi, u_t|_{t=0}=\psi $. In our case the vibration process can be completely described, provided we know both the position and the speed of the string at some $t_0 \in\mathbb {R}$. We will show that certain choices of $t_0$ are suitable for solving some control and observability problems (including also some mixed problems for the semi-infinite $x\in[0,\infty )$ string). In the second part of the paper a minimal restriction on the right-hand side function $f(x,t)$ is presented that guarantees the property $v(x,t)\in C^2(\mathbb{R}^2)$ for the solution $v$ of the problem $$ v_{tt}-a^2 v_{xx}=f(x,t), \quad v|_{t=t_0}=v_t|_{t=t_0}=0, $$ namely $f\in C(\mathbb{R}^2)$ and $f_t \in C(\mathbb{R}^2)$.

DOI: 10.14232/actasm-014-052-5

AMS Subject Classification (1991): 35L05, 35A09, 35Q93

Keyword(s): string vibrations, classical solutions, observation problems, smoothness of the solutions

Received July 3, 2014, and in revised form January 12, 2015. (Registered under 52/2014.)