Abstract
In this work we establish the local solvability of quasilinearsymmetric hyperbolic system using local monotonicity method andfrequency truncation method. The existence of an optimal control isalso proved as an application of these methods.
Highlights
Quasilinear symmetric or symmetrizable hyperbolic systems arise in a wide range of problems in engineering and physics
We establish the solvability of such system using two different methods, viz. local monotonicity method, which was first used in [14] to establish the solvability of stochastic Navier-Stokes equations, and a frequency truncation method ([13], [3])
The new methods we present here are motivated by applications to control theory and stochastic analysis
Summary
Quasilinear symmetric or symmetrizable hyperbolic systems arise in a wide range of problems in engineering and physics. We formulate a simple optimal control problem and demonstrate the utility of the new methods in proving the existence of optimal control. A (t, u)u is locally monotone, a local in time existence and uniqueness of smooth solutions of (1.1) is obtained, using a generalization of the Minty-Browder technique. By considering a truncated quasilinear symmetric hyperbolic syatem, the local solvability of (1.1) is established in section 4 using the fact that the frequency truncated sequence of solutions is Cauchy. As an application of both of these methods, the existence of an optimal control is obtained in section 5 for a typical control problem. There exists a unique solution u(·) of (1.1) with u ∈ C(0, T ∗; Hs(Rn)) ∩ C1(0, T ∗; Hs−1(Rn)) under the following conditions on the operator A (·, ·):. T ∗ < T is the maximal time for which the left hand side of (1.4) is finite
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