These lectures will introduce the audience to the problems of
a-priori, inverse-type inequalities, such as they arise in the Control
Theory of Partial Differential Equations for hyperbolic and
Petrowski-type PDEs, defined on a multi-dimensional bounded domain.
These include:
(i) continuous observability inequalities; (ii) stabilization inequalities. As a consequence, one obtains exact controllability results with control acting on a portion of the boundary; energy decay (stabilization) results with dissipation active on a portion of the boundary; and, a-fortiori, global uniqueness theorems for over-determined problems.
Linear and semi-linear PDEs to be considered include: second order
hyperbolic equations, first order hyperbolic systems; Schrodinger
equations; various plate-like equations with finite or infinite speed
of propagation; linear and semi-linear models; etc, depending on
time-constraints.
More specific topics will include:
(a) Carleman-type inequalities and from here a-priori continuous
observability/ stabilization inequalities for linear equations: from
canonical models to general models with variable coefficients and
energy-level terms. The linear estimates are derived under essentially
minimal assumptions imposed on the regularity of the coefficients, an
within an essentially self-contained approach. This is a critical
starting point for the non-linear analysis of the related control
problems, to be described in (b) and (c) below.
(b) Exact controllability for semi-linear waves, plates equations.
(c) Uniform stability and energy decay estimates for hyperbolic and
Petrowski problems (waves, plates, and even shells if time permits..)
References
R. Triggiani and P.F. Yao,
Carleman estimates with no lower-order terms for general Riemann wave equations. Global uniqueness and observability in one shot R. Gulliver, I. Lasiecka, W. Littman and R. Triggiani, The case for differential geometry in the control of single and coupled PDEs: the structural acoustic chamber I. Lasiecka, R. Triggiani and P.F. Yao, Inverse/observability estimates for second-order hyperbolic equations with variable coefficients I. Lasiecka, R. Triggiani and X. Zhang Nonconservative wave equations with unobserved Neumann b.c.: global uniqueness and observability in one shot R. Triggiani Uniform boundary stabilization of semi-linear wave equations with non-linear boundary damping |