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An observability for parabolic equations from a measurable set in time
Kim Dang Phung, Gengsheng Wang
TL;DR
Observability estimates for parabolic equations are studied when observations are limited to an open spatial subset and a positive-measure subset of time. The paper establishes such an estimate using quantitative unique continuation at one time, and gives applications to bang-bang properties in norm- and time-optimal control.
Problem
The paper studies whether the observability estimate holds when the time observation set is measurable and has positive measure, rather than being the full interval.
Source: §1 · p. 2Method
The estimate is established using Hölder continuous dependence from one point in time, for parabolic equations with space-time-dependent potentials.
Source: §1 · pp. 2–3Results
For measurable E ⊂ (0,T) of positive measure and nonempty open ω ⊂ Ω, Theorem 1.1 establishes an observability estimate for solutions of the parabolic equation.
Source: §1 · p. 2Takeaways & Limitations
The estimate is applied to bang-bang properties for norm- and time-optimal control problems.
Source: §1 · p. 3The time-optimal application may have an empty admissible set PM; conditions on the potentials are needed to guarantee PM is nonempty for some T > 0.
Source: §3.2 · p. 17
Abstract
from arXiv · showhide
This paper presents a new observability estimate for parabolic equations in $\Omega\times(0,T)$, where $\Omega$ is a convex domain. The observation region is restricted over a product set of an open nonempty subset of $\Omega$ and a subset of positive measure in $(0,T)$. This estimate is derived with the aid of a quantitative unique continuation at one point in time. Applications to the bang-bang property for norm and time optimal control problems are provided.
1 Introduction and main result
The paper studies observability when measurements are restricted to an open spatial subset and a positive-measure time set. It establishes the estimate using quantitative unique continuation at one time, for parabolic equations with space-time-dependent potentials.
1 Introduction and main result
- The equation is posed on a bounded, convex, open domain, with coefficients b ∈ L∞(Ω × (0,T))^n and a ∈ L∞(0,T; Lq(Ω)), where q ≥ 2 for n = 1 and q > n for n ≥ 2.For every initial value in L2(Ω), it has a unique solution in C([0,T]; L2(Ω)) that depends continuously on the initial value.Source: §1 · pp. 1–2
- When E is a single time point, the observability estimate does not hold; a Hölder-type quantitative unique-continuation estimate at one time is available instead.For E equal to the full interval or a subinterval, related observability estimates were known for broad classes of potentials.Source: §1 · p. 2
- For measurable E ⊂ (0,T) of positive measure and nonempty open ω ⊂ Ω, Theorem 1.1 gives an observability estimate for solutions of the parabolic equation.The observation region is D = ω × E, and the estimate relates the solution at T to its observation over D.Source: §1 · p. 2
- The proof strategy uses Hölder continuous dependence from one point in time to establish the measurable-time observability estimate.The approach applies to space-time-dependent potentials a and b and differs from the approach in for the zero-potential case; the one-time estimate uses techniques from, and.Source: §1 · pp. 2–3
- The paper also presents applications of Theorem 1.1 to bang-bang properties in norm- and time-optimal control problems.The proof of Theorem 1.1 is developed after the one-time dependence estimate.Source: §1 · p. 3
2 Proof of Theorem 1.1
The proof reduces Theorem 1.1 to two preliminary results, then uses a sequence based at a density point of the measurable observation times. It establishes the estimates in Proposition 2.2 through a separate proof built around auxiliary quantities and intermediate estimates.
2.1 Preliminary results
- The proof of Theorem 1.1 relies on two preliminary results, with one proved in the appendix and the other in subsection 2.3.Source: §2.1 · p. 3
2.2 Proof of Theorem 1.1
- A positive-measure time set supplies a density point and a sequence used in the proof, which applies Proposition 2.2 on successive intervals and sums the resulting estimates.Source: §2.1 · p. 3, §2.2 · pp. 4–5
2.3 Proof of Proposition 2.2
- The Proposition 2.2 proof introduces Gλ and Nλ,ϕ and derives intermediate estimates leading to (2.1.3) and (2.1.4).Source: §2.3 · pp. 6, 13
3 Applications to bang-bang controls
The section applies the observability result to norm- and time-optimal control problems, establishing bang-bang properties and uniqueness of the optimal control when the time-optimal feasible set is nonempty.
3 Applications to bang-bang controls
- For the control system, the section assumes bounded coefficients a and B, bounded divB, and initial data y0 in L2(Ω).The control acts on a nonempty open subset ω of Ω.Source: §3 · p. 14
3.1 Norm optimal bang-bang control
- Every norm-optimal control is unique and has L2(Ω) norm equal to the minimum L∞(τ,T;L2(Ω)) cost for almost every t in (τ,T).The minimum is taken over controls that drive the solution from y0 to zero at time T.Source: §3.1 · p. 15
3.2 Time optimal bang-bang control
- When the time-optimal feasible set PM is nonempty, any time-optimal control has L2(Ω) norm M for almost every t in (τ*,T), and at most one optimal pair exists.The bang-bang property is also used to establish uniqueness of the pair.Source: §3.2 · p. 17
- The time-optimal feasible set may be empty; the paper gives potential conditions that ensure it is nonempty for some T>0.The stated conditions involve a and divB.Source: §3.2 · p. 17
- The time-optimal problem seeks to delay control activation while still driving the solution from y0 to zero by the fixed ending time T.A pair (τ,g) is feasible when g is inactive on Ω×(0,τ) and reaches the zero target at T.Source: §3.2 · p. 17
Proof of Proposition 2.1
The proof selects a density point of the positive-measure set E and constructs nearby times to obtain the stated estimate.
Proof of Proposition 2.1
- Since E has positive measure, almost every point of E is a density point, and the proof chooses such a point ℓ in (0,T).The chosen point provides the center for the subsequent interval construction.Source: Proof of Proposition 2.1 · p. 19
- The construction uses the density estimate near ℓ to bound the lengths of successive intervals and derive the estimate.The interval-measure bounds are combined with the preceding relations to obtain (2.1.2).Source: Proof of Proposition 2.1 · pp. 19–20
- The proof selects ℓ1 just above ℓ and defines a sequence of times ℓm that remains between ℓ and ℓ1.The sequence stays within the radius θo used in the construction.Source: Proof of Proposition 2.1 · p. 19
Some inequalities
This section states coefficient assumptions and uses Hölder and Sobolev inequalities in estimates involving a, φ, and Gλ.
Some inequalities
- The coefficient a is assumed to belong to L∞(0,T;Lq(Ω)), with q≥2 for n=1 and q>n for n≥2.Source: Some inequalities · p. 20
- For each ε > 0, there exists a positive constant C(ε,Ω,n,q) for any φ ∈ H1_0(Ω) and almost every t ∈ (0,L) ⊂ [0,T].Source: Some inequalities · p. 20
- The displayed estimates invoke Hölder and Sobolev inequalities to bound terms involving a and ∇(φ²Gλ).One bound uses ∥a∥L1(Ω) and ∥φ²Gλ∥L∞(Ω).Source: Some inequalities · pp. 20–22