(pp. 13-34)

In this chapter we introduce the method of*backstepping*and present nominal designs that will serve as the basis for all the designs in the rest of the book.

We consider the following class of linear parabolic partial integro-differential equations (P(I)DEs):

$\begin{gathered} {u_t}(x,t) = \varepsilon (x){u_{xx}}(x,t) + b(x){u_x}(x,t) + \lambda (x)u(x,t) \\+ g(x)u(0,t) + \int_0^x {f(x,y)u(y,t)dy} \\ \end{gathered} $(2.1)

for$x \in (0,L)$,$t > 0$, with boundary conditions

${u_x}(0,t) = - qu(0,t)$(2.2)

$u(L,t) = U(t)$, (2.3)

where$U(t)$is the control input. We make the following assumptions:

$\varepsilon (x) > 0{\text{forall}}x \in [0,L],\varepsilon \in {C^3}[0,L],q \in \mathbb{R}$,

$b \in {C^2}[0,L],\lambda ,g \in {C^1}[0,L],f \in {C^1}([0,L] \times [0,L])$. (2.4)

For large positive*q*,$\lambda \left( x \right),$or$f\left( {x,y} \right),$

the plant (2.1) with$U\left( t \right) = 0$is unstable; therefore, the objective is to stabilize he equilibrium u(x, t) ≡ 0.

Before we proceed with the control design, a...