Optimal Control of Partial Differential Equations

10 CHAPTER 1. EXAMPLES OF CONTROL PROBLEMS
1.3.2 Cooling process in metallurgy
In modern steel casting machine, the design of cooling process leads to rather challenging
control problems. The problem is described by a fully nonlinear heat equation of the form
ρ(T )c(T ) ∂T
∂t
= div(k(T )∇T ), in Ω × (0, tf),
where T is the temperature in the domain Ω, c(T ) the specific heat capacity, ρ(T ) the density,
and k(T ) the conductivity of the steel at the temperature T . The heat extraction is ensured
by water sprays corresponding to nonlinear boundary conditions:
k(T ) ∂T
∂n
= R(T, u), on Γ × (0, tf),
where u is the control variable, and the radiation law R is a nonlinear function (for example,
R can be the Stefan-Boltzmann radiation law). The cost functional can be of the form:
J(T, u) = β1 (T (tf) − ¯ T ) 2 tf
+ β2 |u| q
Ω
with β1 > 0, β2 > 0, tf is the terminal time of the process, ¯ T is a desired profile of temperature,
the exponent q is chosen in function of the radiation law R.
In industrial applications, constraints must be added on the temperature. In that case we
shall speak of state constraints. For more details and other examples we refer to [21].
1.4 Control of hyperbolic equations
The control of acoustic noise, the stabilization of flexible structures, the identification of
acoustic sources can be formulated as control problems for hyperbolic equations. Lot of models
have been studied in the literature [16]. The one dimensional models cover the elementary
theories of elastic beam motion [28]. Let us present a stabilization problem for the Timoshenko
model. The equation of motion of the Timoshenko beam is described by the following set of
equations:
ρ ∂2 2 u ∂ u ∂φ
− K −
∂t2 ∂x2 ∂x
∂
Iρ
2φ ∂t2 − EI ∂2φ ∂x
2 + K
φ − ∂u
∂x
= 0, in (0, L),
0
= 0, in (0, L),
where u is the deflection of the beam, φ is the angle of rotation of the beam cross-sections due
to bending. The coefficient ρ is the mass density per unit length, EI is the flexural rigidity
of the beam, Iρ is the mass moment of inertia of the beam cross section, and K is the shear
modulus. If the beam is clamped at x = 0, the corresponding boundary conditions are
u(0, t) = 0 and φ(0, t) = 0 for t ≥ 0.
If a boundary control force f1 and a boundary control moment f2 are applied at x = L, the
boundary conditions are
K(φ(L, t) − ux(L, t)) = f1(t) and − EIφx(L, t) = f2(t) for t ≥ 0.