*2.4. Example*

The following example is a modification of ([20], Example) and ([16], Example 1).

Let Ω ⊂ R*n* with *n* ≥ 4 be a domain with smooth boundary *∂*Ω. Define *ϕ*(*r*) = |*r*|*<sup>γ</sup>*−1*r* for *r* = 0 and 0 < *γ* < *n* − 2 *n*. We consider the following system:

$$\begin{cases} \boldsymbol{\mu}\_{t} \in \Delta \boldsymbol{\varrho}(\boldsymbol{\mu}) + \boldsymbol{G}(t, \boldsymbol{y}, \boldsymbol{\mu}) \\ -\frac{\partial \boldsymbol{\varrho}(\boldsymbol{\mu})}{\partial \boldsymbol{\nu}} \in \boldsymbol{\beta}(\boldsymbol{\mu}) \text{ on } (0, T) \times \partial \Omega \\ \boldsymbol{\mu}(0, \boldsymbol{y}) = \boldsymbol{\mu}\_{0}(\boldsymbol{y}). \end{cases}$$

Here, *u* ∈ R, *∂ϕ*(*u*) *∂ν* is the outward normal derivative on *∂*Ω and *β*(·) is a maximal monotone graph in R with *β*(0) ! 0. The multifunction *G* has nonempty compact values, is measurable on all variables and continuous on the third one.

Define the operator *B* in *L*<sup>1</sup>(Ω) by

$$B\mathfrak{u} = \Delta\mathfrak{q}(\mathfrak{u}), \text{ for } \mathfrak{u} \in D(B), \text{ where}$$

$$D(B) = \{ u \in L^1(\Omega) \colon \varrho(u) \in \mathcal{W}^{1,1}(\Omega), \ \Delta\varrho(u) \in L^1(\Omega), \ -\frac{\partial\varrho(u)}{\partial\nu} \in \beta(u) \text{ on } \partial\Omega\}.$$

The derivatives here are understood in the sense of distributions.

As it is shown in ([4], p. 97), the operator *B* defined above is m-dissipative in *L*<sup>1</sup>(Ω) and generates a noncompact semigroup. Notice that in [4] the author works with m-accretive operators *A*; however *A* is m-dissipative iff −*A* is m-accretive.

Let

$$F(t, \mathbf{x}) = \{ f \in L^1(\Omega) \colon f(y) \in G(t, y, \mathbf{x}(t, y)) \text{ a.e. in } \Omega \},$$

which is jointly measurable and continuous on *x*. We assume also that there exists *h* ∈ *L*<sup>1</sup>([0, *T*]) such that *F*(*<sup>t</sup>*, *x*) ≤ *h*(*t*)(1 + |*x*|). Let *x*0 = *<sup>u</sup>*(·) ∈ *<sup>D</sup>*(*B*). Therefore (H1), (H2) hold true.

Suppose also that there exists a Perron function *<sup>w</sup>*(·, ·) such that for every *x*, *z* ∈ Ω and every *f* ∈ *<sup>F</sup>*(*<sup>t</sup>*, *x*) there exists *g* ∈ *<sup>F</sup>*(*<sup>t</sup>*, *z*) such that

$$\int\_{\Omega^{+}(x\to z)} (f(y) - g(y)) dy - \int\_{\Omega^{-}(x\to z)} (f(y) - g(y)) dy$$

$$\pm \int\_{\Omega^{0}(x\to z)} (f(y) - g(y)) dy \le w \left( t, \int\_{\Omega} |f(y) - g(y)| \right) dy.$$

Here, <sup>Ω</sup>+(<sup>−</sup>,<sup>0</sup>) *x*→*y* = {*y* ∈ Ω; *f*(*y*) > *g*(*y*)(<sup>&</sup>lt;, =)}. It follows from the characterization of [·, ·]+ (see, e.g., [21], Example 1.4.3) that (H3) also hold true.

In the case when *γ* > *n* − 2 *n* the operator *B* generates a compact semigroup and it is of complete continuous type.

#### *2.5. Applications to Optimal Control*

Our results can be applied to the following optimal control problem:

$$\min \left\{ \mathbf{g}(\mathbf{x}(T)) + \int\_{t\_0}^{T} f(t, \mathbf{x}(t)) dt \right\},\tag{16}$$

where *<sup>x</sup>*(·) is a solution of (1). Here, *f*(·, ·) is Carathéodory and integrally bounded on the bounded sets and the function *g* : *X* → R is assumed to be lower semicontinuous.

Assume (H1)–(H3) and (A). In this case, the limit solution set of (1) is compact and moreover, the set of integral solutions of (1) is dense in the set of limit solutions (see Theorem 6 and Lemma 4).

Clearly, in general, the problem (16) has no optimal solution.

**Theorem 7.** *Under the above conditions, the problem* (16) *admits an optimal limit solution.*

**Proof.** The functional *<sup>x</sup>*(·) → *Tt*0 *f*(*<sup>t</sup>*, *x*(*t*))*dt* is continuous from *<sup>C</sup>*(*<sup>I</sup>*, *X*) into R. Furthermore, *<sup>x</sup>*(·) → *g*(*x*(*T*)) is lower semicontinuous. Consequently, the functional *J*(*x*(·)) = *g*(*x*(*T*)) + *Tt*0 *f*(*<sup>t</sup>*, *x*(*t*))*dt* is lower semicontinuous from *<sup>C</sup>*(*<sup>I</sup>*, *X*) into R. The proof follows from the facts that the limit solution set is *<sup>C</sup>*(*<sup>I</sup>*, *X*) compact and every lower semicontinuous real valued function attains its minimum on a compact set.
