Motion planning for the heat equation

Abstract

The paper gives an explicit open‐loop control, able to approximately steer the one‐dimensional heat equation with control on the boundary from any state to any other state. The control is obtained thanks to a parametrization of the solutions of the heat equation by a series involving infinitely many derivatives of the system ‘flat output’. Copyright © 2000 John Wiley & Sons, Ltd.

Control of a Quantum Particle in a Moving Potential Well

“…We give explicit formulae for the control [0, T ] t →v(t), steering in finite time from Ψ = 0, v =v = 0 at t = 0 to Ψ = 0, v = a,v = 0 at t = T for any T > 0. Computations are similar to those we have proposed for heat or Euler-Bernouilli dynamics where ultra-distributions and Gevrey functions of order ≤ 1 appear (Laroche et al, 2000;Fliess et al, 1996;Rouchon, 2001).…”

Control of a Quantum Particle in a Moving Potential Well

“…We give explicit formulae for the control [0, T ] t →v(t), steering in finite time from Ψ = 0, v =v = 0 at t = 0 to Ψ = 0, v = a,v = 0 at t = T for any T > 0. Computations are similar to those we have proposed for heat or Euler-Bernouilli dynamics where ultra-distributions and Gevrey functions of order ≤ 1 appear (Laroche et al, 2000;Fliess et al, 1996;Rouchon, 2001).…”

Flatness-based feedforward control design for flexible structures

“…4. Thereby, the determined feedforward controls u * (t) are applied to the full order FE model (13). Obviously, highly accurate trajectory tracking is achieved for all scenarios as is confirmed by comparing the obtained physical system output w(t) to its respective desired trajectory w * (t).…”

Feedforward tracking control for non‐uniform Timoshenko beam models: combining differential flatness, modal analysis, and FEM

“…y * (t) ∈ C ∞ . The temporal behavior of the respective modal states x * k (t) directly follows from (23), which yields the corresponding trajectory w * (t) for the system output w(t) by evaluation of (20). This observation relies on the convergence of the operators (23), (24), which is verified in the next section.…”

Control of a Quantum Particle in a Moving Potential Well

“…We give explicit formulae for the control [0, T ] t →v(t), steering in finite time from Ψ = 0, v =v = 0 at t = 0 to Ψ = 0, v = a,v = 0 at t = T for any T > 0. Computations are similar to those we have proposed for heat or Euler-Bernouilli dynamics where ultra-distributions and Gevrey functions of order ≤ 1 appear (Laroche et al, 2000;Fliess et al, 1996;Rouchon, 2001).…”

Flatness-based feedforward control design for flexible structures

“…4. Thereby, the determined feedforward controls u * (t) are applied to the full order FE model (13). Obviously, highly accurate trajectory tracking is achieved for all scenarios as is confirmed by comparing the obtained physical system output w(t) to its respective desired trajectory w * (t).…”

Feedforward tracking control for non‐uniform Timoshenko beam models: combining differential flatness, modal analysis, and FEM

“…y * (t) ∈ C ∞ . The temporal behavior of the respective modal states x * k (t) directly follows from (23), which yields the corresponding trajectory w * (t) for the system output w(t) by evaluation of (20). This observation relies on the convergence of the operators (23), (24), which is verified in the next section.…”

Control of a Quantum Particle in a Moving Potential Well

“…We give explicit formulae for the control [0, T ] t →v(t), steering in finite time from Ψ = 0, v =v = 0 at t = 0 to Ψ = 0, v = a,v = 0 at t = T for any T > 0. Computations are similar to those we have proposed for heat or Euler-Bernouilli dynamics where ultra-distributions and Gevrey functions of order ≤ 1 appear (Laroche et al, 2000;Fliess et al, 1996;Rouchon, 2001).…”

Flatness-based feedforward control design for flexible structures

“…4. Thereby, the determined feedforward controls u * (t) are applied to the full order FE model (13). Obviously, highly accurate trajectory tracking is achieved for all scenarios as is confirmed by comparing the obtained physical system output w(t) to its respective desired trajectory w * (t).…”

Feedforward tracking control for non‐uniform Timoshenko beam models: combining differential flatness, modal analysis, and FEM

“…y * (t) ∈ C ∞ . The temporal behavior of the respective modal states x * k (t) directly follows from (23), which yields the corresponding trajectory w * (t) for the system output w(t) by evaluation of (20). This observation relies on the convergence of the operators (23), (24), which is verified in the next section.…”