[Paper Review] Observability inequalities from measurable sets for some evolution equations

Gengsheng Wang, Can Zhang|arXiv (Cornell University)|Jun 13, 2014

Stability and Controllability of Differential Equations30 references6 citations

TL;DR

This paper establishes observability inequalities from measurable time sets for abstract evolution equations in Hilbert spaces, using two distinct settings: analytic semigroups with admissible observation operators and $C_0$ semigroups with spectral-like conditions. The key contribution is proving that observability from measurable sets implies the bang-bang property for time-optimal control problems in parabolic equations and related systems.

ABSTRACT

In this paper, we build up two observability inequalities from measurable sets in time for some evolution equations in Hilbert spaces from two different settings. The equation reads: $u'=Au,\; t>0$, and the observation operator is denoted by $B$. In the first setting, we assume that $A$ generates an analytic semigroup, $B$ is an admissible observation operator for this semigroup (cf. \cite{TG}), and the pair $(A,B)$ verifies some observability inequality from time intervals. With the help of the propagation estimate of analytic functions (cf. \cite{V}) and a telescoping series method provided in the current paper, we establish an observability inequality from measurable sets in time. In the second setting, we suppose that $A$ generates a $C_0$ semigroup, $B$ is a linear and bounded operator, and the pair $(A, B)$ verifies some spectral-like condition. With the aid of methods developed in \cite{AEWZ} and \cite{PW2} respectively, we first obtain an interpolation inequality at one time, and then derive an observability inequality from measurable sets in time. These two observability inequalities are applied to get the bang-bang property for some time optimal control problems.

Motivation & Objective

• To establish observability inequalities from measurable subsets of time for abstract evolution equations in Hilbert spaces.
• To extend existing observability results—previously limited to specific PDEs like the heat equation—to general evolution equations governed by semigroups.
• To apply these inequalities to prove the bang-bang property for time-optimal control problems in parabolic-type systems.
• To unify and generalize prior results on observability and control by introducing new techniques applicable to both analytic and $C_0$ semigroup settings.

Proposed method

• Uses propagation estimates of analytic functions to control the growth of solutions in the first setting involving analytic semigroups.
• Applies a novel telescoping series method to bridge local observability estimates into global inequalities over measurable sets.
• Employs interpolation inequalities derived from spectral-like conditions in the second setting, relying on subspaces with exponential decay properties.
• Implements methods from [2] and [29] to derive pointwise observability estimates under Hypothesis (H), which involves spectral projection and decay rates.
• Establishes bounds on the solution norm at time $T$ using $L^1$-type integrals of observation data over measurable sets $E \subset (0,T)$.
• Applies duality arguments and adjoint semigroup analysis to transfer observability results to control problems, particularly for the time-optimal control setting.

Experimental results

Research questions

• RQ1 Can observability inequalities be established from measurable time sets rather than intervals for abstract evolution equations?
• RQ2 What conditions on the generator $A$ and observation operator $B$ ensure such observability from measurable sets?
• RQ3 How can the observability inequality from measurable sets be used to deduce structural properties of time-optimal controls?
• RQ4 What is the role of analytic semigroups and spectral conditions in enabling observability from sets of positive measure?
• RQ5 To what extent do the derived inequalities imply the bang-bang property in time-optimal control problems?

Key findings

• The paper proves that for analytic semigroups with admissible observation operators satisfying a local observability inequality, observability holds from any measurable subset $E \subset (0,T)$ with positive measure.
• An explicit constant $C$ in the observability inequality (1.4) depends on $E$, $T$, $d$, $k$, and $\|B\|_{\mathcal{L}(D(A),U)}$, ensuring stability under measurable observation sets.
• For $C_0$ semigroups under Hypothesis (H), a pointwise interpolation inequality (1.5) is derived, linking the solution norm at time $t \in (0,1]$ to the observation norm and initial data.
• The main inequality (1.6) shows that $\|S(T)u_0\|_X$ is bounded by an $L^1$-type integral of $\|BS(t)u_0\|_U$ over any measurable $E \subset (0,T)$ of positive measure.
• The results are applied to prove the bang-bang property for time-optimal control problems in the Grushin operator system and parabolic equations with jumping coefficients.
• Corollaries 3.6 and 3.7 confirm the bang-bang property for time-optimal control of the Grushin-type equation and parabolic equations with discontinuous coefficients, respectively.