** Download Code** related to the Numerical aspects of LTHC of Burgers equation.

**Developed by Navid Allahverdi, , Alejandro Pozo & Enrique Zuazua**

#### Turnpike property for functionals involving L^{1}−norm

** Download Code** related to the Turnpike property for functionals involving L^{1}−norm.

**Developed by Martin Lazar, Cesare Molinari & Enrique Zuazua**

#### Turnpike property for functionals involving L^{1}−norm

We want to study the following optimal control problem:

\begin{equation*}

\left(\mathcal{P}\right) \ \ \ \ \ \ \ \hat{u}\in\argmin_{u\in L^2_T} \left\{J\left(u\right)=\alpha_c \norm{u}_{1,T} + \frac{\beta}{2}\norm{u}^2_{T}+\alpha_s \norm{Lu}_{1,T} + \frac{\gamma}{2}\norm{Lu-z}_{T}^2\right\},

\end{equation*}

#### Numerical aspects of LTHC of Burgers equation

This issue is motivated by the challenging problem of sonic-boom minimization for supersonic aircrafts, which is governed by a Burgers-like equation. The travel time of the signal to the ground is larger than the time scale of the initial disturbance by orders of magnitude and this motivates our study of large time control of the sonic-boom propagation…

#### Long time control and the Turnpike property

The turnpike property establishes that, when a general optimal control problem is settled in large time, for most of the time the optimal control and trajectories remain exponentially close to the optimal control and state of the corresponding steady-state or static optimal control problem…

#### Long time horizon control (LTHC)

Control problems for evolution PDEs are most often considered in finite time intervals, without paying attention to the length of the control horizon and how it affects the nature of optimal trajectories and controls. However, the effective available time horizon is one of the critical factors in applications, as it occurs in the design of…