Discretization-Optimization Methods for Nonlinear Elliptic Optimal Control Problems with State Constraints

Discretization-Optimization Methods for Nonlinear Elliptic Optimal Control Problems with State Constraints

I. Chryssoverghi¹, J. Geiser², J. Al-Hawasy¹

(¹) Department of Mathematics, School of Applied Mathematics and Physics National Technical University of Athens (NTUA)

Zografou Campus, 15780 Athens, Greece e-mail: ichris@central.ntua.gr

(²) Weierstrass Institute for Applied Analysis and Stochastics (WIAS) Mohrenstrasse 39, D-10117 Berlin, Germany

e-mail: geiser@wias-berlin.de

Abstract

We consider an optimal control problem described by a second order elliptic boundary value problem, jointly nonlinear in the state and control, with control and state constraints, where the state constraints and cost functionals involve also the state gradient. Since this problem may have no classical solutions, it is also formulated in the relaxed form. The classical problem is discretized by using a finite element method for state approximation, while the controls are approximated by elementwise constant, or linear, or multilinear, controls. Various necessary conditions for optimality are given for the classical and the relaxed problem, in the continuous and the discrete case. We then study the behavior in the limit of discrete optimality, and of discrete extremality and admissibility. Next, we apply a penalized gradient projection method to each discrete problem, and also a progressively refining version of this method to the continuous classical problem. We prove that accumulation points of sequences generated by the first method are extremal for the discrete problem, and that strong classical (resp. relaxed) accumulation points of sequences of discrete controls generated by the second method are admissible and weakly extremal classical (resp. relaxed) for the continuous classical (resp. relaxed) problem.

Finally, numerical examples are given.

Keywords. Optimal control, nonlinear elliptic systems, state constraints, discretization, finite elements, discrete penalized gradient projection method, progressive refining.

1 Introduction

We consider an optimal control problem described by a second order elliptic boundary value problem, which is jointly nonlinear in the state and control, with control and state constraints, where the state constraints and cost functionals involve also the gradient of the state. The problem is discretized by using a Galerkin finite element method with continuous elementwise linear basis functions for state approximation, while the controls are approximated by (not necessarily continuous) elementwise constant, or linear, or multilinear, controls. Various necessary conditions for optimality are given for the classical and the relaxed problem, in the continuous and the discrete case. Under appropriate assumptions, we prove that strong accumulation points in of sequences of optimal (resp. admissible and extremal) discrete controls are optimal (resp. admissible and weakly extremal classical) for the continuous classical problem, and that relaxed accumulation points of sequences of optimal (resp. admissible and extremal) discrete controls are optimal (resp. admissible and weakly extremal relaxed) for the continuous relaxed problem. We then apply a penalized gradient projection method to each discrete problem, and also a corresponding discrete method to the continuous classical problem, which progressively refines the discretization during the iterations, thus reducing computing

L2

time and memory. We prove that accumulation points of sequences generated by the first method are extremal for the discrete problem, and that strong classical (resp.

relaxed) accumulation points of sequences of discrete controls generated by the second method are admissible and weakly extremal classical (resp. relaxed) for the continuous classical (resp. relaxed) problem. Finally, numerical examples are given.

For approximation and optimization methods applied to distributed optimal control problems, see e.g. [2], [5,6], [8-12], [16-18], and the references therein.

2. The continuous optimal control problems

Let be a bounded domain in , with Lipschitz boundary Γ. Consider the nonlinear elliptic state equation

Ω \^d

(2.1) Ay+ f x y x w x( , ( ), ( )) 0= in Ω, (2.2) y x( ) 0= on , Γ

where is the formal second order elliptic differential operator A (2.3)

1 1

: ^{d d} ( / _i)[ ( )_ij / _j

j i

Ay x a x y x

= =

= −

∑∑

∂ ∂ ∂ ∂ ^].

The constraints on the control are ( )w x ∈U in Ω, where U is a compact subset of

\ν, the state constraints are

(2.4) G w_m( ) : g_m( , ( ),x y x y x w x dx( ), ( )) 0, m p

=

∫

Ω ∇ = ^{=1,..., ,}

(2.5) G w_m( ) : g_m( , ( ),x y x y x w x dx( ), ( )) 0,

=

∫

Ω ∇ ≤ ^{m= +p 1,..., ,q}

, and the cost functional is

(2.6) G w₀( ) : g x y x₀( , ( ), y x w x dx( ), ( )) .

=

∫

Ω ∇

The state equation will be interpreted in the following weak form

(2.7) y V∈ , and a y v( , ) b x y x w x v x dx( , ( ), ( )) ( ) f x w x v x dx( , ( )) ( ) ,

Ω Ω

+

∫

=

∫

^{∀ ∈v V}

where a( , )⋅ ⋅ is the usual bilinear form associated with and defined on A V V× (2.8)

, 1

( , ) : ^d _ij( ) .

i j i j

y v

a y v a x dx

x x

= Ω

= ∂ ∂

∑ ∫

∂ ∂

Defining the set of classical controls

(2.9) W: { := w x6w x w( ) measurable from Ω to }U ⊂L^∞( )Ω ⊂L²( )Ω ,

the continuous classical optimal control problem is to minimize subject to and to the above state constraints.

P G₀

w W∈

It is well known that such nonconvex optimal control problems may have no classical solutions, but reformulated in the so-called relaxed form, they have a solution in an extended space under weak assumptions. Next, we define the set of relaxed controls (or Young measures; for the relevant theory, see [19], [15])

(2.10) R: { := r Ω →M U₁( ) r weakly measurable}⊂L^∞_w( ,Ω M U( ))≡L¹( , ( ))*Ω C U , where ( )M U (resp. M U₁( )) is the set of Radon (resp. probability) measures on U. The set R is endowed with the relative weak star topology, and R is convex, metrizable and compact. If each classical control ( )w ⋅ is identified with its associated Dirac relaxed control r( ) :⋅ =δ_{w( )⋅} , then W may also be considered as a subset of R,

and W is thus dense in R. For a given φ∈L¹( ; ( ))Ω C U =L¹( ; ( ))Ω C U (or ( , ; )

B U

φ∈ Ω \ , where ( , ; )B ΩU \ is the set of Caratheodory functions in the sense of Warga [19]) and r∈L^∞_w( ,Ω M U( )) (in particular, for r∈R), we shall use the notation (2.11) ( , ( )) : ( , ) ( )( ),

x r x U x u r x du

φ =

∫

φ

and ( , ( ))φ x r x is thus linear (under convex combinations, for ) in . A sequence ( converges to r in

r∈R r

k)

r ∈R R iff

(2.12) lim ( , ( ))_k ( , ( ))

k φ x r x dx φ x r x dx

Ω Ω

→∞

∫

=

∫

^,

for every φ∈L¹( ; ( ))Ω C U , or φ∈ ΩB( , ; )U \ , or φ∈ Ω×C( U).

We denote by ⋅ the Euclidean norm in \ⁿ, by ⋅ _∞ the norm in , by and

( , ⁿ) L^∞ Ω \ ( , )⋅ ⋅ ⋅ the inner product and norm in L²( ;Ω \ⁿ), and by ( , )⋅ ⋅ ₁ and ⋅ ₁ the inner product and norm in the Sobolev space V:=H₀¹(Ω). We can now formulate the relaxed problem as follows. The relaxed state equation (in weak form) is given by (2.13) y V∈ and a y v( , ) f x y x r x v x dx( , ( ), ( )) ( ) 0,

+

∫

Ω = ^{∀ ∈v V,}

the control constraint is r∈R, and the relaxed functionals are (2.14) G r_m( ) : g_m( , ( ),x y x y x r x dx( ), ( )) ,

=

∫

Ω ∇ ^{m=0,... .q}

The continuous relaxed optimal control Problem P is to minimize subject to the constraints

0( ) G r (2.15) r∈R, G r_m( ) 0,= m=1,..., ,p G r_m( ) 0,≤ 1,..., .m= +p q

In the sequel, we shall make some of the following assumptions.

Assumptions 2.1 The coefficients a_ij satisfy the ellipticity condition

(2.16) ₀ ²

, 1 1

( ) ,

d d

ij i j i

i j i

a x z z α z

= =

∑

≥

∑

^{∀z zi, j∈\, x∈Ω,}

with α₀ >0, a_ij∈L^∞( )Ω , which implies that

(2.17) a y v( , ) ≤α₁ y₁ v₁, a v v( , )≥α₂ v₁², ∀y v V, ∈ , for some α₁≥0,α₂>0.

Assumptions 2.2 The functions f and f_y are defined on Ω× ×\ U , measurable for fixed y u, , continuous for fixed x, and satisfy

(2.18) f x( ,0, )u ≤φ₀( ),x ( , )∀ x u ∈Ω×U,

where φ₀∈L^s( )Ω , with s≥2, s≥n/ 2 (e.g. s=2, for n=1, 2,3), and (2.19) 0≤ f_y( , , )x y u ≤φ₁( ) (x η₁ y), ( , , )∀ x y u ∈Ω× ×\ U,

where η₁ is an increasing function from [0,+∞) to [0,+∞), if the functionals depend on , and

1 L ( )

φ ∈ ^∞ Ω Gm ∇y φ₁∈L^s( )Ω otherwise.

Assumptions 2.3 The functions g_m are defined on Ω×\^d+1×U, measurable for fixed , continuous for fixed

, ',

y y u x, and satisfy

(2.20) g_m( , , ', )x y y u ≤ψ_0m( )x +β_0m y' ,² ( , , ', )x y y u ^d+1 U

∀ ∈Ω×\ × with y ≤C',

where C'>C, ψ_0m∈L¹( )Ω , β_0m≥0.

Assumptions 2.4 The functions g_m,g_my,g_my' are defined on Ω×\^d+1×U', where is an open set containing the compact set U, measurable on for fixed

, continuous on

' U Ω

( , ', )y y u ∈\^d+1×U' \^d+1×U' for fixed x∈Ω, and g_my,g_my' satisfy (2.21) g_my( , , ', )x y y u _1m( )x _1m y'^{2( 1)},

ρ

ψ β ρ⁻

≤ +

(2.22) g_my'( , , ', )x y y u ≤ψ_2m( )x +β_2m y' , (2.23) g_mu( , , ', )x y y u ≤ψ_3m( )x +β_3m y' ,

( , , ', )x y y u ^d+1 U'

∀ ∈Ω×\ × , with y ≤C',

where C'<C, ψ_im∈L²( )Ω , 0β_im ≥ , [1, )ρ∈ ∞ if n=1 or 2, 2

: 2

n ρ σ< = n

− if n≥3. The following theorem follows directly form Theorem 3.1 in [3].

Theorem 2.1 Under Assumptions 2.1-2, for every relaxed control , the state equation has a unique solution

r∈R : _r

y = y ∈ ∩V C^α( )Ω , for some α∈(0,1). Moreover, there exists constants ,C C such that

(2.24) y_{r 1}+ y_{r ∞} ≤C, y_{r C}α ≤C, for every r∈R.

The following proposition is a simple generalization of Proposition 2.1 in [7], and will be useful in the sequel. It can be proved by using the (possible) convergence

, the fact that a converging sequence in

rk →r L^s is dominated (in norm a.e. in Ω,

and up to a subsequence) by a fixed function in L^s, Hölder’s inequality, Egorov’s theorem, and Lebesgue’s dominated convergence theorem.

Proposition 2.1 For i=1,...,K, K ≥0, let s_i∈ +∞[1, ], [0, ]σ_i∈ s_i if s_i< +∞, σ_i: 0= if s_i = +∞, with

0 1

1 ^{K i} 1

i i

s s

σ

=

+

∑

≤ ^{, 1 : 0}

si = if s_i = +∞. Let be a function defined on , measurable for every fixed, continuous for every

F ( ^N)^K U

Ω× \ × y u, x fixed, and

satisfying (2.25)

1

( , , ) ( ) ( ) ( )

K

i i

i

F x y u x x ξ y

=

≤ Φ + Ψ

∏

^,

for every ( , , )x y u ∈Ω×(\^N)^K×U, with y_i ≤C_i if s_i = +∞,

where y: ( ,...,= y₁ y_K), Φ ∈L¹( )Ω , Ψ ∈L^s0( )Ω , ξ_i( y_i ) := y_i ^σi if s_i < +∞, ( ) :

i yi

ξ =1 if s_i = +∞. If ( converges to in strongly, , with

)

k

yi y_i L^si( ;Ω \^N) i=1,...,K

k

i i

y ∞ ≤C (for sufficiently large) if k s_i = +∞, and ( )r^k converges to in r R, then

(2.26) lim ( , ^k( ), ( ))^k ( , ( ), ( )) .

k F x y x r x dx F x y x r x dx

Ω Ω

→∞

∫

=

∫

Theorem 2.2 Under Assumptions 2.1-3, the operator r6 y_r (resp. w6 y_w), from R (resp. W with the relative topology of L²( ;Ω \^ν), hence of L^∞( ;Ω \^ν)) to V , and

to C₀( )Ω , and the functionals r 6G r_m( ) on R (resp. on W with the same topologies) are continuous. If the relaxed problem has an admissible control (i.e.

satisfying all the constraints), then it has a solution.

m( ) w6G w

Proof. Let be a sequence that converges to in W, with the relative topology of

( )w_k w

2( , )

L Ω \^ν . Since the corresponding sequence of states is bounded in V and in

( )y_k

0( )

C^α Ω , for some α∈(0,1), and since the injection of C₀^α( )Ω into C₀( )Ω is compact, there exists a subsequence (same notation) converging to some in V weakly and in

y

0( )

C Ω strongly. Let any v V∈ be given. By the state equation (2.27) a y v( , )_k f x y x w x vdx( , ( ),_{k k}( )) 0

+

∫

Ω = ^.

By the mean value theorem and since η₁ is increasing, we have, for every with y y ≤C ( defined in Theorem 2.1), and for some C µ( ) [0,1]x ∈

(2.28) f x y u v( , , ) ≤ f x( ,0, )u v + f x y u v( , , ) − f x( ,0, )u v

0 1 1

( ,0, ) _y( , ( ) , ) ( ) ( ) ( ( ) )

f x u v f x µ x y u yv φ x φ x vη µ x y y

= + ≤ +

0( )x 1( )x v 1( )C C

φ φ η

≤ + ,

Since φ₀∈L^s, φ₁∈L^s or L^∞, v V∈ ⊂L², and y_k →y in L^∞, we can apply Proposition 2.1 to pass to the limit in the state equation for and find that . Next, we have

yk y=y_w

(2.29) ₂ ²

1 ( ,

n n n

y y a y y y y

α − ≤ − − )

y

( ( ,f y w^{n n}),yⁿ) a y y( , ⁿ) a y( ⁿ y y, ) 0,

= − − − − →

since y_k → in V weakly and ( ( ,f y w^{n n}),yⁿ)→( ( , ), )f y w y by Proposition 2.1, which shows that in V strongly. The convergence of the original sequence follows from the uniqueness of the limit. The continuity of the functionals follows then from Proposition 2.1. The proofs for and are similar. The existence of an optimal relaxed control follows from the compactness of

yn →y

Gm

r6 yr G_m:R→\

R and the continuity of the functionals G_m (the set of admissible controls is a closed subset of

R).

Note that the classical problem may have no classical solution, and since , we generally have

W ⊂R

(2.30) _{0 0} ,

constraints on constraints on

: min ( ) inf ( ) :

R W

r w

c = G r ≤ G w =c

where the equality holds, in particular, if there are no state constraints, as W is dense in R. Since usually approximation methods slightly violate the state constraints, approximating an optimal relaxed control by a relaxed or a classical control, hence the possibly lower relaxed optimal cost , is not a drawback in practice (see [19], p.

259).

cR

Lemma 2.1 Under Assumptions 2.1-4, dropping the index m in , , for , the functional G, defined on

gm G_m , '

r r ∈R R (resp. W, with U convex) is l-

differentiable at (resp. ) for every integer l, i.e. for every and any choice of l controls (resp. ),

r w l

ri∈R w_i∈W i=1,...,l, we have

(2.31)

1 1

( ^l _i( _i )) ( ) ^l ( , _i ) _i ( _i

i i

G r r r G r DG r r r o

1

),

l

i

ε ε ε

= =

+

∑

− − =

∑

− +

∑

=

(resp.

1 1

( ^l _i( _i )) ( ) ^l ( , _i ) _i ( _i

i i

G w w w G w DG w w w o

1

)

l

i

ε ε ε

= =

+

∑

− − =

∑

− +

∑

= ^),

for 0,ε_i ≥

1

1,

l i i

ε

=

∑

≤

with DG r r( , _i r) : H x y( , , y z r x, , ( )_i r x dx( )) ,

− =

∫

Ω ∇ −

(resp. DG w w( , _i w) : H x y_u( , , y z w w x, , )( ( )_i w x dx( )) ),

− =

∫

Ω ∇ −

where the Hamiltonian is defined by

(2.32) H x y y z u( , , ', , ) := −z f y x u( , , )+g x y y u( , , ', ),

and the adjoint state z:= ∈z_r V satisfies the linear adjoint equation (2.33) a v z( , ) ( ( , ) , ) ( ( ,+ f_y y r z v = g_y y ∇y r v, ), ) (+ g_y'( ,y ∇y r, ),∇v),

∇ (resp. a v z( , ) ( ( , ) , ) ( ( ,+ f_y y w z v = g_y y ∇y w v, ), ) (+ g_y'( ,y ∇y w, ), v) ),

,

∀ ∈v V with y:= y_r (resp. y:= y_w).

In particular, the directional derivative of the functional G defined on R (resp. W, with U convex) is given by

(2.34)

0

( ( )) (

( , ) limG r r r G r

DG r r r

α

α ) α

→ +

+ − −

− =

( , ( ), ( ), ( ), '( ) ( )) H x y x y x z x r x r x dx

=

∫

Ω ∇ − ^,

(resp.

0

( ( )) (

( , ) lim G r w w G w

DG r w w

α

α ) α

→ +

+ − −

− =

( , ( ), ( ), ( ), ( ))( ( ) ( )) H x y xu y x z x w x w x w x dx

=

∫

Ω ∇ − ^).

Moreover, the operator r6z_r, from R to V (resp. , from W to V ), and the functional

w6zw

( , )r r 6DG r r( , −r), on R R× (resp. ( , )w w 6DG r w w( , − ), on W ), are continuous.

×W Proof. We shall prove the l-differentiability for classical controls only; we could also prove the Fréchet differentiability in this case, but the proof will be thus similar to the proof for relaxed ones. We first remark that, by our assumptions and since the injection V ⊂L^ρ is continuous, the functional

(2.35) v6( ( ,g_y y ∇y w v, ), ) (+ g_y'( ,y ∇y w, ),∇v)

belongs to the dual V* of V , and f_y( , )y w ∈L^s( )Ω , 2≤ ≤ ∞s , . Hence the linear adjoint equation has a unique solution

( , ) 0 fy y w ≥

z∈V, for every w W∈ , by the Lax- Milgram theorem (if s= ∞), or by Lemma 3.2 in [3] (if s< ∞, no in ). Now let

, ,

'

y g

w W∈ w_i∈W ε_i∈(0,1), i=1,...,l, ε: ( ,..., )= ε₁ ε_l , with

1

: ^l _i 1

i

ε ε

=

=

∑

≤ ^{, and set}

(2.36) ,

1

: ^l _i( _i )

i

w_ε w ε w w

=

= +

∑

− ^{δwi:=wi−w, :y = yw, :y}ε = ^ywε , :δ_εy = y_ε −y. From the state equation, we have

(2.37) (a δy v_ε, ) ( ( ,+ f y w_{ε ε})− f y w v( , ), )

( , ) ( ( , ) ( , ), ) ( ( , ) ( , ), ) a δy v_ε f y w_{ε ε} f y w_ε v f y w_ε f y w v 0

= + − + − = .

Using the mean value theorem, we see that δy_ε satisfies the linear equation

(2.38) , ,

1

( , ) ( (_y ) , )) ^l ( _{i u}( , ) , )

i

a δy v_ε f y µδy_ε δy v_ε ε f y w µδw_ε δw v

=

+ + = −

∑

+ ^{i ∀ ∈v V}

where the functions

(2.39) a:= f_y(y+µδy_ε) (with a≥0),

1

: ( , )

l

i u i

i

f ε f y w µδw_ε δw

=

= −

∑

+ ^,

belong to L^∞( )Ω (or L^s) and ( )L^s Ω , respectively, by our assumptions. It then follows from Lemma 3.2 in [3] that

(2.40) _{1 s} ' .

y y c f L c

ε ε

δ + δ _∞ ≤ ≤ ε

Now, by our assumptions, the functional on the open subset Y×L²( ,Ω \^d)×W' of ( ) 2( , ^d) ( , ^v)

L^∞ Ω ×L Ω \ ×L^∞ Ω \

(2.41) ( , ', ) :y y w g x y y w dx( , , ', ) , Φ =

∫

Ω

where

(2.42) ^Y:=

{

^φ∈^{L∞( )}Ω ^φ ∞ <^C'

}

, ^{W' :=}

{

^{ψ∈L∞( )Ω ψ :Ω →U'}

}

^,

has the Fréchet derivative defined by (2.43) '( , , ', )(Φ x y y w δ δy, y',δw)

' .

[ ( , , ', )g_y x y y wδy g_y( , , ', )x y y wδy' g x y y w_u( , , ', ) w dx]

=

∫

Ω + + ^δ

This can be shown under our assumptions by using the mean value theorem in max- form, the Cauchy-Schwartz inequality, and Proposition 2.1. Using then the above estimate on δ_εy, we have

(2.44)

1

( ^l _{i i} )

i

o δ_εy _∞ δ_εy ε δw o( )

= ∞

+ ∇ +

∑

= ^ε

ε

, hence

(2.45) G w( )_ε G w( ) g_y( ,y y w, )δ_εydx g_y'( ,y y w, ) δ ydx

Ω Ω

− =

∫

∇ +

∫

∇ ∇

1

( , , ) ( )

l

i u i

i

g y y w w dx o

ε δ ε

= Ω

+

∑ ∫

∇ + ^.

Similarly, the state equation, for v:=z, yields by linearization (2.46)

1

( , ) ( ( , ) , ) ( ( , ) , ) ( ) 0

l

y i u i

i

a δ_εy z f y w δ_εy z ε f y w δw z o

=

+ +

∑

+ ^{ε = .}

On the other hand, the adjoint equation, for :v =δy_ε, yields

(2.47) a(δ_εy z, ) ( ( , ) ,+ f_y y w z δ_εy) ( ( , ),= g_y y w δ_εy) (+ g_y'( , ),y w ∇δ_εy).

Gathering the above results, we obtain (2.48)

1

( ) ( ) [ ( , ) ( , , )] ( )

l

i u u i

i

G w_ε G w ε z f y w g y y w δw dx o ε

= Ω

− =

∑ ∫

− + ∇ +

1

( , ( ), ( ), ( ), ( )) ( ).

l

i u i

i

H x y x y x z x w x w dx o

ε δ ε

= Ω

=

∑ ∫

∇ ⁺

Finally, the continuity of the operator is proved by using the continuity of , from W to , the compact injection , and Proposition 2.1. The continuity of the functional

w6zw

w6 yw L^∞ V ⊂L²

( , )w w 6DG r w w( , − ) follows from the above continuities. The continuity proofs for relaxed controls are similar.

The following theorem states various continuous necessary conditions for optimality.

Theorem 2.3 Under Assumptions 2.1-4, if r∈R (resp. w W∈ , with U convex) is optimal for Problem P or (resp. Problem ), then (resp. ) is strongly extremal relaxed (resp. weakly extremal classical), i.e. there exist multipliers

P P r w

λm∈\, , with

0,...,

m= q

(2.49) λ₀ ≥0, 0λ_m≥ , m= +p 1,...,q,

0

1

q m m

λ

=

∑

= ^, such that

(2.50)

0

( , ) 0,

q

m m

m

DG r r r λ

=

∑

− ≥ ^{∀ ∈r R,}

(2.51) ( ) 0,λ_mG r_m = m= +p 1,...,q (relaxed transversality conditions).

(resp.

(2.52)

0

( , ) 0,

q

m m

m

DG w w w λ

=

− ≥

∑

^{∀ ∈w W ,}

(2.53) ( ) 0,λ_mG w_m = m= +p 1,...,q (classical transversality conditions) ).

The global condition (2.50) is equivalent to the strong relaxed pointwise minimum principle

(2.54) ( , ( ), ( ), ( ), ( )) min ( , ( ), ( ), ( ), ),

u U

H x y x y x z x r x H x y x y x z x u

∇ = ∈ ∇ a.e. in Ω,

where the complete Hamiltonian and adjoint , are defined with replaced by . If U is convex, then this principle implies the weak relaxed pointwise minimum principle

H z g

0 q

m m

m

λ g

∑

=

(2.55) H x y z r x r x_u( , , , ( )) ( ) minH x y z r x_u( , , , ( )) ( , ( )),x r x

φ φ

= a.e. in Ω

where the minimum is taken over the set ( , ; )B ΩU U of Caratheodory functions (see [18]), which in turn implies the global weak relaxed condition

(2.56) H x y z r x_u( , , , ( ))[ ( , ( ))φ x r x r x dx( )] 0,

Ω − ≥

∫

^{∀ ∈ Ωφ B( , ; )U U .}

A control satisfying this condition and the above transversality conditions is called weakly extremal relaxed. The global condition (2.52) is equivalent to the weak classical pointwise minimum principle

r

(2.57) ( , ( ),_u ( ), ( ), ( )) ( ) min _u( , ( ), ( ), ( ), ( )) , H x y x y x z x w x w x u U H x y x y x z x w x u

∇ = ∈ ∇

a.e. in Ω.

Proof. The functionals G_m, m=0,...,q, defined on R (resp. W) are continuous (Theorem 2.1) and, by Lemma 2.2, (p+1)-differentiable (cost and p equality state constraints) at (resp. ). The global condition (i) (resp. (iii)) and the transversality conditions (ii) (resp. (iv)) follow then from the general multiplier theorem V.2.3 (resp.

V.3.2) in [19] ( depends here on the control only, since or is unique for every or ). The equivalence of the global and pointwise conditions is standard, in both cases (see e.g. [19]) since U is closed (it has a dense denumerable subset). Now, the strong relaxed pointwise minimum principle can be written, for a.a. ,

r w

Gm y_r y_w

r w

x∈Ω x fixed

(2.58) ( ) ( ) ( ), .

UH u r du ≤H u

∫

^{∀ ∈u U}

Let φ∈ ΩB( , ; )U U be any Caratheodory function. Since U is convex here, we have (2.59) ( ) ( ) ( ( ( ) )),

UH u r du ≤H u+ε φ u −u

∫

^{∀ ∈u U, [0,1]∀ ∈ε ,}

hence

(2.60) ( ) ( ) ( ( ( ) )) ( ).

UH u r du ≤ UH u+ε φ u −u r du

∫ ∫

By the Mean Value Theorem and the uniform continuity of H in u

(2.61) ( ( ( ) )) ( )

0 (

U

H u u u H u

r du) ε φ

ε

+ − −

≤

∫

( ( )( ( ) ))( ( ) ) ( )

UH uu εµ u φ u u φ u u r du

=

∫

+ − − ^{( 0≤µ( ) 1u ≤ )}

( )( ( ) ) ( ) ( )

UH uu φ u u r du α ε

=

∫

− ^{+ ,}

where ( )α ε →0 as ε →0, hence

(2.62) _u( )( ( ) ) ( ) ' ( )( ( )_u ) 0,

UH u φ u −u r du =H r φ r −r

∫

^≥

for every φ∈ ΩB( , ; )U U , a.e. in Ω, which is the weak relaxed minimum principle.

By integration, we get the global weak relaxed condition (2.63) H r_u( )( ( )φ r r dxdt) 0,

Ω − ≥

∫

^{∀ ∈ Ωφ B( , ; )U U .}

Remark. In the absence of equality state constraints, it can be shown that if the optimal control is regular, i.e. there exists w w'∈W such that

(2.64) G w_m( )+DG w w_m( , '−w) 0< , m= +p 1,...,q,

(Slater condition), then λ₀ ≠0 for any multipliers as in Theorem 2.3.

3 Discretizations and behavior in the limit

We suppose in Sections 3 and 4 that Ω is a polyhedron (for simplicity). For each integer n≥0, let { }E_i^{n N}_i=1ⁿ be an admissible regular partition of Ω into elements (e.g.

-simplices), with as . Let be the

subspace of functions that are continuous on

d hⁿ =max [diam(_i E_iⁿ)]→0 n→ ∞ Vⁿ ⊂V

Ω and linear (or multilinear) on each element . The set of discrete controls is defined as the subset of (not necessarily continuous) controls that are (optionally) constant, or linear, or multilinear, on each element , and (optionally) such that

n

Ei Wⁿ ⊂W

wn n

Ei wⁿ

∞ L

∇ ≤ , with independent of (this reduces to a finite number of linear constraints on the coefficients defining ). We endow with the Euclidean topology.

L n

wn Wⁿ

Remark. If Ω has an appropriately piecewise C¹ boundary Γ, one can approximate by a polyhedral one , with domain

Γ Γⁿ Ωⁿ, up to ; the results of this section

still hold in this case, with slight modifications in the definitions of and in the proof of Lemma 3.2 (interpolation inside

( )ⁿ o h

, V Wn ⁿ

Ωn and zero values on Γⁿ).

The following assumptions are stronger than Assumptions 2.2-4.

Assumptions 3.1 The functions , ,f f_y f_u (resp ) are defined on ' (resp. on ), with

, , ',

m my my mu

g g g g

Ω× ×\ U Ω×\^d+1×U' U'⊃U open, measurable for fixed (resp. ), continuous for fixed

, y u , ',

y y u x, and satisfy

(3.1) f x y u( , , ) ≤c₁(1+ y^{ρ −1}), (3.2) 0≤ f x y u_y( , , )≤c₂(1+ y^{ρ −2}), (3.3) f x y u_u( , , ) ≤c₃(1+ y^{ρ −1}),

( , , )x y u U',

∀ ∈Ω× ×\

(3.4) g_m( , , ', )x y y u ≤c₄(1+ y^ρ+ y' ),² (3.5) g_my( , , ', )x y y u c₅(1 y ¹ y'^{2( 1)}),

ρ ρ

ρ

− −

≤ + +

(3.6) g_my'( , , ', )x y y u c₆(1 y ² y' ),

≤ + ρ + (3.7) g_mu( , , ', )x y y u c₇(1 y ² y' ),

≤ + ρ + ( , , ', )x y y u ^d+1 U',

∀ ∈Ω×\ ×

where , c_i ≥0 ρ∈ ∞[1, ) if n=1 or 2, 2

: 2

n ρ σ< =n

− if . Note that each of the above inequalities is also satisfied if it holds for some

3 n≥

i 0

c ≥ and ρ∈[1, )ρ .

For a given discrete control , the discrete state is the solution of the discrete state equation

wn∈W^{n n}

n n

v V

: n

n n

y = yw ∈V (3.8) a y v( , ) ( ( ,^{n n} + f y w^{n n}), ) 0,vⁿ = ∀ ∈ .

The following theorem can be proved by using the techniques in [13] (via Brouwer’s fixed point theorem), under our coercivity, monotonicity and continuity assumptions.

Theorem 3.1 Under Assumptions 2.1 and 3.1 (on ,f f_y), for every control , the discrete state equation has a unique solution

n n

w ∈W

n n

y ∈V .

The discrete state equation, which is a nonlinear system, can be solved by iterative methods. The discrete functionals, defined on Wⁿ, are given by

(3.9) G r_mⁿ( )ⁿ g_m( ,yⁿ y w dxⁿ, ⁿ) ,

=

∫

Ω ∇ ^{m=0,..., .q}

The discrete control constraint is and the discrete state constraints are either of the two following ones

wn∈Wⁿ (3.10) Case (a) G w_mⁿ( ⁿ) ≤ε_mⁿ, m=1,..., ,p (3.11) Case (b) G w_mⁿ( ⁿ)=ε_mⁿ, 1,..., ,m= p and

(3.12) (G w_m^{n n})≤ε_mⁿ, ε_mⁿ ≥0, m= +p 1,..., ,q

where the feasibility perturbations ε_mⁿ are chosen numbers converging to zero, to be defined later. The discrete relaxed optimal control Problem P_aⁿ (resp. P_bⁿ) is to

minimize subject to and the above state constraints, Case (a) (resp.

Case (b)).

( )

n n

G wm wⁿ∈Wⁿ

The proof of the following theorem parallels that of Theorem 2.1, noting that all norms are equivalent in the finite dimensional space Vⁿ.

Theorem 3.2 Under Assumptions 2.1 and 3.1 (on ,f f_y), the operator , from to , are continuous. Under assumptions 2.1 and 3.1 (on

wn 6 yⁿ

Wn Vⁿ f f, ,_y g_m), the

functionals , on , are continuous, and for every n, if Problem , or , is feasible, then it has a solution.

( )

n n

w 6G wm ⁿ Wⁿ P_aⁿ

n

Pb

The proofs of the following lemma and theorem also parallel the continuous case.

Lemma 3.1 Under Assumptions 2.1 and 3.1, dropping in the functionals, is l- differentiable for every l, and its directional derivative is given for

m Gⁿ

,

n n

w w ∈Wⁿ by (3.13) DG w wⁿ( ⁿ, ⁿ wⁿ) H x y_u( , ⁿ, y z wⁿ, ,^{n n})(wⁿ w dxⁿ)

− =

∫

Ω ∇ − ^,

n

∇ zn

where the discrete adjoint state satisfies the linear discrete adjoint equation

: ⁿ

n n

z =z w ∈V

(3.14) a z v( , ) (^{n n} + z fⁿ _y( ,y w^{n n}), ) ( ( ,vⁿ = g_y yⁿ ∇y wⁿ, ⁿ), ) (vⁿ + g_y'( ,yⁿ ∇y wⁿ, ⁿ), vⁿ), ,

n n

v V

∀ ∈ where yⁿ:=yⁿwⁿ.

Moreover, the operator wⁿ 6 , from Wⁿ to Vⁿ, and the functional (w wⁿ, ⁿ)6DG w wⁿ( ⁿ, ⁿ−wⁿ), on Wⁿ×Wⁿ, are continuous.

n

q

Theorem 3.3 Under Assumptions 2.1 and 3.1, if is optimal for Problem , then is weakly discrete extremal classical (or discrete extremal), i.e. there exist

multipliers , , with , ,

wn∈W P_bⁿ

wn

n

λm∈\ m=0,..., λ_mⁿ ≥0 λ_mⁿ ≥0 m= +p 1,...,q,

0

1

q n m m

λ

=

∑

= ^, such that

(3.15)

0

( , ) ( , , , ) 0,

q

n n n n n n n n n n n

m m

m

DG w w w H y y z w w dx

λ Ω

=

− = ∇ − ≥

∑ ∫

^{∀ ∈wn Wn,}

(3.16) (λ_mⁿ G w_m( ⁿ)−ε_mⁿ) 0,= m= +p 1,..., ,q

where and are defined with replaced by . The global condition (3.17) is equivalent to the strong discrete classical elementwise minimum principle

Hn zⁿ g

0 q

n

m m

m

λ g

∑

=

(3.18) _n ( , , , ) min _n ( , , , )

i i

n n n n n n n n n n n

u u

E H y y z w w dx u U E H y y z w udx

∇ = ∈ ∇

∫ ∫

^{, i=1,...,Nn.}

Let Wⁿ be the set of elementwise constant discrete controls. Clearly, Wn ⊂Wⁿ in all cases. The following control approximation result (i) (resp. (ii)) is proved similarly to the corresponding result in [8] (resp. [13]).

Proposition 3.2 (i) For every r∈R, there exists a sequence (wⁿ∈Wⁿ) of discrete classical controls, considered as relaxed ones, that converges to in r R.

(ii) For every w W∈ , there exists a sequence (wⁿ∈Wⁿ) of discrete classical controls, considered as relaxed ones, that converges to in w L² strongly.

The following key lemma gives consistency results.

Lemma 3.2 We suppose that Assumptions 2.1 and 3.1 are satisfied and drop m in the functionals.

(i) If the sequence (wⁿ∈Wⁿ) converges to r∈R in R (resp. to in

strongly), then (resp. ) in V strongly, (resp.

), and (resp. ) in

w W∈ L²

n

y → yr

z z →

n

y →yw G wⁿ( ⁿ)→G r( )

( ) ( )

n n

G w →G w zⁿ → _r ⁿ z_w L^ρ( )Ω strongly (and in V strongly, if the functionals do not depend on ∇y).

(ii) If the sequences (wⁿ∈Wⁿ) and (wⁿ∈Wⁿ) converge to and w w, respectively, in W, then

(3.19) DG w wⁿ( ⁿ, ⁿ −wⁿ)→DG w w w( , − ).

Proof. (i) Suppose that wⁿ →r in R. From the discrete state equation, we have (3.20) a y( ,ⁿ yⁿ) ( ( ,+ f y w^{n n}) ( (0,− f wⁿ),yⁿ− = −0) ( (0,f wⁿ),yⁿ),

and since f is increasing in y (3.21) ₂ ²

1 ( , ) ( (0, ), ) (0, ) 1,

n n n n n n n

y a y y f w y f w s y c y

α ≤ ≤ ≤ ≤ ⁿ

which shows that the sequence is bounded in V . By Alaoglu’s theorem, there exists a subsequence (same notation) that converges weakly in V to some , and since the injection of V in

( )yⁿ

y∈V ( )

L^ρ Ω is compact (see Ref. 20), we can suppose that in strongly. For any given

yn →y L^ρ( )Ω v∈C₀¹( )Ω , let Vⁿ be the sequence of interpolates of v at the vertices of the partition of

(vⁿ∈ )

Ω. This sequence converges to in

v C¹₀( )Ω (hence in V ) strongly. We have (3.22) a y v( , ) ( ( ,^{n n} + f y w^{n n}), ) 0.vⁿ =

Since wⁿ →r in R and in V strongly, hence in strongly, by Proposition 2.1 and our assumptions, we can pass to the limit in this equation and find

yn → y L^ρ( )Ω

(3.23) a y v( , ) ( ( , ), ) 0,+ f y r v =

which holds also for every v V∈ ⊂L^s, as C₀¹( )Ω is dense in V . Therefore . The convergence in strongly of the initial sequence follows then from the uniqueness of the limit. Next, we have

y= yr

( ) L^ρ Ω

(3.24) ₂ ²

1 ( , ) ( ( , ), ) ( , ) ( ,

n n n n n n n n

y y a y y y y f y w y a y y a y y y

α − ≤ − − = − − − − ).

y

By Proposition 2.1 and the above convergences of , the last expression converges to zero; hence in V strongly. The convergence follows from the above convergences and the same proposition. From the adjoint equation, we have

( )yⁿ

yn → G wⁿ( ⁿ)→G r( )

(3.25) ₂ ²

1 ( , ) ( ( , ) , )

n n n n n n

z a z z fy y r z z

α ≤ + ⁿ

( ( ,g_y yⁿ y z rⁿ, , ), )^{n n} zⁿ (g_y'( ,yⁿ y z rⁿ, , ),^{n n} zⁿ)

≤ ∇ + ∇ ∇

2( 1)

1 2

4 5

( (1 ), ) ( (1 ), )

p p

n p n p n n n

c y y z c y y z

− −

≤ + + ∇ + + + ∇ ⁿ