Esteve C, Zuazua E (2020)

**Publication Type:** Journal article

**Publication year:** 2020

**Book Volume:** 52

**Pages Range:** 5627-5657

**Journal Issue:** 6

**DOI:** 10.1137/20M1330130

We study the inverse problem, or inverse design problem, for a time-evolution Hamilton-Jacobi equation. More precisely, given a target function uT and a time horizon T > 0, we aim to construct all the initial conditions for which the viscosity solution coincides with uT at time T. As is common in this kind of nonlinear equation, the target might not be reachable. We first study the existence of at least one initial condition leading the system to the given target. The natural candidate, which indeed allows determining the reachability of uT , is the one obtained by reversing the direction of time in the equation, considering uT as terminal condition. In this case, we use the notion of backward viscosity solution, which provides existence and uniqueness for the terminal-value problem. We also give an equivalent reachability condition based on a differential inequality, which relates the reachability of the target with its semiconcavity properties. Then, for the case when uT is reachable, we construct the set of all the initial conditions for which the viscosity solution coincides with uT at time T. Note that, in general, such initial conditions are not unique. Finally, for the case when the target uT is not necessarily reachable, we study the projection of uT on the set of reachable targets, obtained by solving the problem backward and then forward in time. This projection is then identified with the solution of a fully nonlinear obstacle problem and can be interpreted as the semiconcave envelope of uT , i.e., the smallest reachable target bounded from below by uT.

Enrique Zuazua Iriondo
Lehrstuhl für Dynamics, Control, Machine Learning and Numerics (Alexander von Humboldt-Professur)

**APA:**

Esteve, C., & Zuazua, E. (2020). The inverse problem for Hamilton-jacobi equations and semiconcave envelopes. *SIAM Journal on Mathematical Analysis*, *52*(6), 5627-5657. https://doi.org/10.1137/20M1330130

**MLA:**

Esteve, Carlos, and Enrique Zuazua. "The inverse problem for Hamilton-jacobi equations and semiconcave envelopes." *SIAM Journal on Mathematical Analysis* 52.6 (2020): 5627-5657.

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