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

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

I. Chryssoverghi¹, I. Coletsos¹, J. Geiser², B. Kokkinis¹

(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 semilinear parabolic partial differential equation, with control and state constraints, where the state constraints and cost involve also the state gradient. Since this problem may have no classical solutions, it is reformulated in the relaxed form. The relaxed control problem is discretized by using a finite element method in space involving numerical integration and an implicit theta-scheme in time for space approximation, while the controls are approximated by blockwise constant relaxed controls. Under appropriate assumptions, we prove that relaxed accumulation points of sequences of optimal (resp. admissible and extremal) discrete relaxed controls are optimal (resp. admissible and extremal) for the continuous relaxed problem. We then apply a penalized conditional descent method to each discrete problem, and also a progressively refining version of this method to the continuous relaxed problem. We prove that accumulation points of sequences generated by the first method are extremal for the discrete problem, and that relaxed accumulation points of sequences of discrete controls generated by the second method are admissible and extremal for the continuous relaxed problem. Finally, numerical examples are given.

Keywords. Optimal control, semilinear parabolic systems, state constraints, relaxed controls, discretization, θ-scheme, discrete penalized conditional descent method.

AMS Subject Classification: 49M25, 49M05, 65N30.

1 Introduction

We consider an optimal distributed control problem for systems governed by a semilinear parabolic partial differential equation, with control and state constraints, where the state constraints and cost involve also the gradient of the state. The problem is motivated, for example, by the control of a heat (or other) diffusion process whose source is nonlinear in the heat and temperature, with nonconvex cost and control constraint set (e.g. on-off type control). Since this problem may have no classical solutions, it is reformulated in the relaxed form, using Young measures. The relaxed problem is discretized by using a Galerkin finite element method with continuous piecewise linear basis functions in space and an implicit theta-scheme in time for space approximation, while the controls are approximated by blockwise constant Young measures. We first state the necessary conditions for optimality for the continuous problems, and then for the discrete relaxed problem. Under appropriate assumptions, we prove that relaxed accumulation points of sequences of optimal (resp. admissible and extremal) discrete relaxed controls are optimal (resp. admissible and extremal) for the continuous relaxed problem. We then apply a penalized conditional descent method to each discrete problem, which generates Gamkrelidze

controls, and also a corresponding discretization-optimization method to the continuous relaxed problem that progressively refines the discretization during the iterations, thus reducing computing time and memory. We prove that accumulation points of sequences generated by the fixed discretization method are extremal for the discrete problem, and that relaxed accumulation points of sequences of discrete controls generated by the progressively refining method are admissible and extremal for the continuous relaxed problem. Using a standard procedure, the computed Gamkrelidze controls can then be approximated by piecewise constant classical ones.

Finally, numerical examples are given. For approximation of nonconvex optimal control and variational problems, and of Young measures, see e.g. [2-9], [12-14] and the references there.

2 The continuous optimal control problems

Let Ω be a bounded domain in \^d, with boundary Γ, and let I =(0, )T , , be an interval. Consider the semilinear parabolic state equation

T < ∞ (2.1) y_t +A t y( ) +a x t₀( , )^T∇ +y b x t y x t w x t( , , ( , ), ( , ))= f x t y x t w x t( , , ( , ), ( , )) (2.2) in Q= Ω ×I,

(2.3) y x t( , ) 0= in Σ = Γ×I, y x( ,0)= y x⁰( ) in Ω,

where A t( ) is the formal second order elliptic differential operator (2.4)

1 1

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

j i

A t y x a x t y x

= =

= −

∑∑

∂ ∂ ∂ ∂

The constraints on the control are ( , )w x t ∈U in where U is a compact, not necessarily convex, subset of , the state constraints are

, Q

'

\d

(2.5) _m( ) : _m( , , , , ) 0,

G w =

∫

Qg x t y ∇y w dxdt= ^{m=1,..., ,p} (2.6) _m( ) : _m( , , , , ) 0,

G w =

∫

Qg x t y ∇y w dxdt≤ ^{m= +p 1,..., ,q} and the cost functional to be minimized is

(2.7) ₀( ) ₀( , , , , ) . G w =

∫

Qg x t y ∇y w dxdt Defining the set of classical controls

(2.8) W: { : ( , )= w x t 6w x t w( , ) measurable from Q to },U

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

0( ) G w w W∈

Next, we define the set of relaxed controls (Young measures; for the relevant theory, see [20], [17])

(2.9) R: { := r Q→M U₁( ) r weakly measurable}⊂L Q M U^∞_w( , ( ))≡L Q C U¹( , ( ))*, 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 of L Q C U¹( , ( ))*. The set R is convex, metrizable and compact. If we identify every classical control with its associated Dirac relaxed control

( ) w ⋅ ( ) _w( )

r ⋅ =δ _⋅ , then W may be considered as a subset of R, and W is thus dense in R. For φ∈L Q C U¹( , ( ))=L Q C U¹( , ( )) (or

( , ; ) B Q U

φ∈ \ , where ( , ; )B Q U \ is the set of Caratheodory functions in the sense of Warga [20]) and r∈L Q M U^∞_w( , ( )) (in particular, for r∈R), we shall use the notation (2.10) ( , , ( , )) : ( , , ) ( , )( ),

x t r x t U x t u r x t du

φ =

∫

φ

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

∈ r

k)

r ∈R R iff

(2.11) lim ( , , ( , ))_k ( , , ( , ))

Q Q

k φ x t r x t dxdt φ x t r x t dxdt

→∞

∫

=

∫

^,

for every φ∈L Q C U¹( ; ( )), or φ∈B Q U( , ; )\ , or φ∈C Q U( × ).

We denote by ⋅ the Euclidean norm in \ⁿ, by ( , )⋅ ⋅ and ⋅ the inner product and norm in L²( )Ω , by ( , )⋅ ⋅ _Q and ⋅ _Q the inner product and norm in , by and

2( ) L Q ( , )⋅ ⋅1 ⋅ ₁ the inner product and norm in the Sobolev space , and by the duality bracket between the dual

1

: 0( V =H Ω) )

< ⋅ ⋅ >, V*=H⁻¹(Ω and V . We also define the usual bilinear form associated with A t( ) and defined on V V×

(2.12)

1 1

( , , ) : ^{d d} _ij( , ) .

j i i j

y v

a t y v a x t dx

x x

= = Ω

= ∂ ∂

∑∑∫

∂ ∂

The relaxed formulation of the above control problem is the following. The relaxed state equation, interpreted in weak form, is

(2.13) < y v_t, > +a t y v( , , ) ( ( )+ a t₀ ^T∇y v, ) ( ( , , ), ) ( ( , , ), ),+ b t y w v = f t y w v ,

∀ ∈v V a.e. in I,

(2.14) y t( )∈V a.e. in I , y(0)= y⁰,

(the derivative is understood here in the sense of V -vector valued distributions), the control constraint is , and the state constraints and cost functionals are

yt

r∈R

(2.15) _m( ) : _m( , , , , ( , )) ,

G r =

∫

Qg x t y ∇y r x t dxdt ^{m=0,..., .q}

The continuous relaxed optimal control problem is to minimize subject to the constraints

0( ) G r (2.16) 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 Γ is Lipschitz if b=0; else, Γ is C¹ and n≤3. Assumptions 2.2 The coefficients a_ij satisfy the ellipticity condition

(2.17) ₀ ²

1 1 1

( , ) ,

d d d

ij i j i

j i i

a x t z z α z

= = =

∑∑

≥

∑

^{∀z zi, j∈\, ( , )x t ∈Q,}

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

(2.18) a t y v( , , ) ≤α₁ y ₁ v ₁, a t v v( , , )≥α₂ v ₁², ∀y v V, ∈ , t∈I, for some α₁≥0,α₂>0.

Assumptions 2.3 a₀∈L Q^∞( )^d, and the functions b,f are defined on measurable for fixed , continuous for fixed

, Q× ×\ U ,

y u x t, , and satisfy the conditions (2.19) b x t y u( , , , ) ≤φ( , )x t +β y², b x t y u y( , , , ) ≥0,

(2.20) f x t y u( , , , ) ≤ψ( , )x t +γ y ,

( , , , )x t y u Q U,

∀ ∈ ×\×

(2.21) f x t y u( , , , )₁ − f x t y u( , , , )₂ ≤L y₁− y₂ , ∀( , , , , )x t y y u_{1 2} ∈ ×Q \²×U, (2.22) b x t y u( , , , )₁ ≤b x t y u( , , , )₂ , ∀( , , , , )x t y y u_{1 2} ∈ ×Q \²×U , with y₁ ≤y₂, where φ ψ, ∈L Q²( ), β γ, ≥0

U .

Assumptions 2.4 The functions g_m are defined on Q×\^d+1× , measurable for fixed , continuous for fixed

,

y u x t, , and satisfy

(2.23) g_m( , , , , )x t y y u ≤ζ_m( , )x t +δ_my²+δ_m y², ∀( , , , , )x t y y u ∈ ×Q \^d+1×U, with ζ_m∈L Q¹( ), δ_m≥0, δ_m≥0.

Assumptions 2.5 The functions b b, ,_y f_y (resp. g_m,g_my,g_my) are defined on Q U

(resp. ), measurable on for fixed ( (resp.

× ×\

1

Q×\d⁺ ×U Q y u, )∈ ×\ U

( , , )y y u ∈\^d+1×U ) and continuous on \×U (resp. \^d+1×U ) for fixed ( , )x t ∈Q, and satisfy

(2.24) b x t y u_y( , , , ) ≤ξ( , )x t +η y, f_y( , , , )x t y u ≤L₁, ( , , , )x t y u Q U

∀ ∈ ×\× ,

(2.25) g_my( , , , , )x t y y u ≤ζ_m1( , )x t +δ_m1 y +δ_m1 y , (2.26) g_my( , , , , )x t y y u ≤ζ_m2( , )x t +δ_m2 y +δ_m2 y ,

( , , , , )x t y y u Q ^d+1 U,

∀ ∈ ×\ ×

m m L Q

ξ ζ ζ ∈

with , ₁, ₂ ²( ), η δ δ δ δ, _m1, _m1, _m2, _m2 ≥0.

The following theorem can be proved by monotonicity and compactness arguments (for continuous b f, and y⁰∈V, see also a proof contained in Theorem 3.1 and Lemma 4.2 below).

Theorem 2.1 Under Assumptions 2.1-3, for every control r∈R and (or ), the relaxed state equation has a unique solution such that

, . Moreover, is essentially equal to a function in

0 2( )

y ∈L Ω

y0∈V y:=y_r

2( , )

y∈L I V y_t∈L I V²( , *) y ( , ( ))2

C I L Ω , and thus the initial condition is well defined.

The following lemma and theorem can be proved by using the techniques of [5], [7], [17].

Lemma 2.1 Under Assumptions 2.1-3, the operator r6 y_r, from R to , and to if . Under Assumptions 2.1-4, the functionals ,

, from

2( , ) L I V

2( , ( ))4

L I L Ω b≠0 r6G r_m( )

0,...,

m= q R to , are continuous. \

It is well known that, even if the control set U is convex, the classical problem may have no classical solutions. But we have anyway the following theorem stating the existence of an optimal relaxed control.

Theorem 2.2 Under Assumptions 2.1-4, if there exists an admissible control (i.e.

satisfying all the constraints), then there exists an optimal relaxed control.

Since W ⊂R, we generally have

(2.27) _{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 [20], p.

259).

cR

The following lemma and theorem can be proved by using the techniques of [5], [7], [20] (see also [11]).

Lemma 2.2 Under Assumptions 2.1-5, dropping the index in the functionals, the directional derivative of is given, for

m G r r, '∈R, by

(2.28)

0

( ( ' )) (

( , ' ) : lim G r r r G r DG r r r

ε

ε ) ε

→ +

+ − −

− =

( , , , , , '( , ) ( , )) ,

QH x t y y z r x t r x t dxdt

=

∫

∇ −

where the Hamiltonian H is defined by

(2.29) H x t y y z u( , , , , , ) :=z f x t y u[ ( , , , )−b x t y u( , , , )]+g x t y y u( , , , , ), and the adjoint state z=z_r satisfies the linear adjoint equation (2.30) − <z v_t, > +a t v z( , , ) (+ a₀^T∇v z, ) (+ zb y r v_y( , ), )

(zf_y( , )y r g_y( , ), )y r v ( ( ,g_y y y r, ), v),

= + + ∇ ∇ ∀ ∈v V, a.e. in I,

(2.31) z t( )∈V a.e. in I, z T( ) 0= ,

with y= y_r. The mappings r6z_r, from R to , and ( , , from

2( )

L Q r r')6DG r r( , '−r) R×R to , are continuous. \

Next, we state the relaxed necessary conditions for optimality.

Theorem 2.3 Under Assumptions 2.1-5, if r∈R is optimal for either the relaxed or the classical optimal control problem, then is extremal, i.e. there exist multipliers r

λm∈\, m=0,...,q, with λ₀ ≥0, 0λ_m≥ , 1,...,m= +p q,

0

1

q m m

λ

=

∑

= , such that

(2.32)

0

( , ' ) 0,

q

m m

m

DG r r r λ

=

∑

− ≥ ^{∀ ∈r' R,}

(2.33) ( ) 0,λ_mG r_m = m= +p 1,...,q (transversality conditions).

The global condition (2.32) is equivalent to the strong relaxed pointwise minimum principle

(2.34) ( , , ( , ), ( , ), ( , ), ( , )) min ( , , ( , ), ( , ), ( , ), ),

u U

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

∇ = ∈ ∇

a.e. in Q,

where complete Hamiltonian and adjoint H z, are defined with .

0

:

q

m m

m

g λ g

=

=

∑

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

(2.35) G r_m( )+DG r r_m( , '− <r) 0, m= +p 1,...,q,

(Slater condition), then λ₀ ≠0 for any set of multipliers as in Theorem 2.2.

3 The discrete optimal control problems

Assumptions 3.1 Γ is appropriately piecewise C¹ if b=0, Γ is and if , is independent of t (for simplicity) and symmetric if

C1 n≤3 0

b≠ a t y v( , , ) θ ≠1 in the θ-

scheme below, the functions a b b b₀, , ,_{y u},f f, ,_y f_u, are continuous (possibly finitely piecewise in t) on the closure of their domains of definition, and

.

0, , , ,_y a b b f f_y y0∈V

Under Assumptions 3.1, for each integer n≥0, let Ωⁿ be a subdomain of Ω with polyhedral boundary such that , an admissible regular quasi-uniform triangulation of

Γn dist( , )Γ Γ =ⁿ o h( )ⁿ { }E_i^{n M}_i=1ⁿ

Ωn into closed d-simplices (elements), with as n , and

max [diam( )] 0

n n

i i

h = E → → ∞ { }Iⁿ_j ^N_j=ⁿ₁, a subdivision of the interval I into closed intervals Iⁿ_j =[tⁿ_j−1,tⁿ_j], of equal length ∆tⁿ, with ∆ →tⁿ 0 as n . We define the blocks . Let be the subspace of functions that are continuous on

→ ∞

n n

ij i j

Q =E ×Iⁿ Vⁿ ⊂V

Ω, are linear (i.e. affine) on each E_iⁿ, and vanish on Ω − Ωⁿ. Let be any given fixed point in U. The set of discrete classical controls is the subset of classical controls that are constant on the interior of each block and equal to on

u0

Wn ⊂W

n

Qij

u0 Q− ×Ω(I ⁿ). The set of discrete relaxed controls Rⁿ ⊂R is the subset of relaxed controls that are equal to a constant measure in M U₁( ) on the interior of each block Q_ijⁿ and equal to δ_u0 on Q− ×Ω(I ⁿ). The set Rⁿ is endowed with the relative weak star topology of M U( )^MN. Clearly, we have . For implementation reasons, one could alternatively use a coarser partition for the discrete controls, that is, use discrete relaxed controls that are constant on hyperblocks

, where the are appropriate unions of some elements and Wn ⊂Rⁿ

'ⁿ

' ' ' '

'ⁿ_{i j} 'ⁿ_{i j}

Q =E ×I E'_iⁿ_' E_iⁿ I'ⁿ_j'

are appropriate unions of some intervals Iⁿ_j.

For a given discrete control rⁿ∈Rⁿ, and θ∈[1/ 2,1] if b=0, θ =1 if , the corresponding discrete state is given by the discrete state equation (implicit

0 b≠ : ( ,...,0 )

n n n

y = y yN

θ-scheme)

(3.1) (1/∆tⁿ)(yⁿ_j −yⁿ_j−1, )v +a y( ⁿ_jθ, ) (v + a₀^T( )tⁿ_jθ ∇y_jθ, ) ( ( ,v + b tⁿ_jθ yⁿ_jθ, ), )r_jⁿ v ( ( ,f tⁿ_jθ yⁿ_jθ, ), ),r_jⁿ v

= for every v V∈ ⁿ, j=1,..., ,N

(3.2) (y₀ⁿ −y v⁰, )₁=0, for every v V∈ ⁿ, yⁿ_j∈Vⁿ, j=1,...,N, where we set

(3.3) yⁿ_jθ : (1= −θ)yⁿ_j−1+θyⁿ_j, tⁿ_jθ : (1= −θ)tⁿ_j−1+θtⁿ_j.

Theorem 3.1 Under Assumptions 2.2-3 and 3.1, if ∆ ≤tⁿ c' (resp. ), for some ' sufficiently small, independent of and , if

'( )2

tn c h

∆ ≤ ⁿ

c n rⁿ b=0 (resp. b≠0), then, for

every and every control , the discrete state equation has a unique solution such that

n rⁿ yⁿ

n

yj ≤c, 0,...,j= N , with independent of c n and . rⁿ

Proof (sketch). Suppose first that b≠0, θ =1. Lemma 4.1 below shows that, if the solution yⁿ_j exists for every j, then yⁿ_j ≤c, 0,...,j= N , with c independent of and , for as in the assumptions. Suppose by induction that exists for

. Then the solution is a fixed point of the mapping

n

rn ∆tⁿ y_kⁿ

1

k≤ −j yⁿ_j z=F y_θ( ) (here with

θ =1), where is the solution, for given, of the equations z y

(3.4) (1/∆tⁿ)(z−yⁿ_j−1, )v +a(θz+ −(1 θ)yⁿ_j−1, ) (v + a₀^T( ) (tⁿ_jθ ∇θy+ −(1 θ)yⁿ_j−1), )v )

n n n n n

j j j j j

b t_θ θy θ y ₋ r v f t_θ θy θ y ₋ r v

+ + − = + − ∀ ∈v Vⁿ

4

1 1

( ( , (1 ) , ), ) ( ( , (1 ) , ), ,_jⁿ , which reduce (choosing a basis in ) to a regular linear system in . We then show (using our assumptions, the continuous injection , and the inverse inequality, see [10]) that

Vn z

1

H0⊂L ( ) 2

z = F y_θ ≤ c, if y ≤2c, for ∆tⁿ as above, i.e. F_θ maps the closed ball (0, 2 )B c of center 0 and radius 2 in into itself. Moreover, one can see (using also the mean value theorem for b) that

c Vⁿ

F_θ is also contractive in this ball, for ∆tⁿ as above. Therefore F_θ has a unique fixed point in (0, 2 )B c , which is the solution yⁿ_j. If b=0, [1/ 2,1]θ∈ , one can easily see that, by the Lipschitz continuity of f in , the mapping y F_θ is contractive on the whole space , for sufficiently small; hence

Vn ∆tⁿ F_θ has a unique fixed point in Vⁿ.

The solution can be computed by the predictor-corrector method, using the linearized semi-implicit predictor scheme, i.e. with or .

n

yj

0

: ( 1) (0, 2

n

j j

y =F y_{θ −} ∈B c)

n

Vn

The discrete control constraint is rⁿ∈R and the discrete functionals are

(3.5) m q

1

0

( ) : ( , , , ) ,

N

n n n n n n n

m m j j j j

j

G r t g t_θ y_θ y_θ r dx

−

= Ω

= ∆

∑∫

∇ ^{=0,..., .}

The discrete state constraints are either of the two following ones (3.6) Case (a) G r_mⁿ( )ⁿ ≤ε_mⁿ, m=1,..., ,p

(3.7) Case (b) G r_mⁿ( )ⁿ =ε_mⁿ, m=1,..., ,p and

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

where the feasibility perturbations ε_mⁿ are given numbers converging to zero, to be defined later. The discrete cost functional to be minimized is G r₀ⁿ( )ⁿ .

Theorem 3.2 Under Assumptions 2.2-4 and 3.1, the mappings and , defined on

n

r 6 yⁿj

( )n

n n

r 6G rm Rⁿ, are continuous. If any of the discrete problems is feasible, then it has a solution.

Proof. The continuity of the operators rⁿ 6 yⁿ_j is easily proved by induction on j (or by using the discrete Bellman-Gronwall inequality, see [18]). The continuity of

( )

n n

r →G rm ⁿ follows from the continuity of . The existence of an optimal control follows then from the compactness of

gm

Rn.

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

Lemma 3.2 We drop the index m in and . Under Assumptions 2.2-5 and 3.1, for , the directional derivative of the functional is given by

gm G_mⁿ , '

n n

r r ∈Rⁿ Gⁿ

(3.9) ¹ _,1

0

( , ' ) ^N ( , , , , ' ) ,

n n n n n n n n n n n

j j j j j j

j

DG r r r t H t_θ y_θ y_θ z _θ r r dx

− Ω −

=

− = ∆

∑∫

∇ −

where the discrete adjoint zⁿ is given by the linear adjoint scheme

(3.10) −(1/∆tⁿ)(zⁿ_j −zⁿ_j−1, )v +a v z( , ⁿ_j,1−θ) (+ a₀^T∇v z, ⁿ_j,1−θ) (+ zⁿ_j,1−θb t_y( ,ⁿ_jθ yⁿ_jθ, ), )r_jⁿ v (zⁿ_j,1_−θ f t_y( ,ⁿ_jθ yⁿ_jθ, )r_jⁿ g t_y( ,ⁿ_jθ yⁿ_jθ, yⁿ_jθ, ), ) ( ( ,r_jⁿ v g t_y ⁿ_jθ yⁿ_jθ, yⁿ_jθ, ),r_jⁿ v)

= + ∇ + ∇ ∇ ,

, v Vn

∀ ∈ j=N,...,1, z_Nⁿ =0, zⁿ_j∈Vⁿ,

which has a unique solution zⁿ_j−1 for each j, for ∆tⁿ sufficiently small. Moreover, the mappings rⁿ 6zⁿ and are continuous.

n

( , ' )r r^{n n} 6DG r rⁿ( , '^{n n}−rⁿ)

Theorem 3.3 Under Assumptions 2.2-5 and 3.1, if rⁿ∈R is optimal for the discrete problem with state constraints, Case (b), then it is extremal, i.e. there exist multipliers

, , with

n

λm∈\ m=0,...,q

(3.11) , , λ_mⁿ ≥0 λ_mⁿ ≥0 m= +p 1,...,q,

0

1

q n m m

λ

=

∑

= ^, such that

(3.12) _,1

0 1

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

q N

n n n n n n n n n n n n n

m m j j j j j j

m j

DG r r r t H t_θ y_θ y_θ z _θ r r dx

λ ₋

= = Ω

− = ∆ ∇ − ≥

∑ ∑∫

'^{n n},

r R

∀ ∈

(3.13) [λ_mⁿ G r_m( )ⁿ −ε_mⁿ] 0,= m= +p 1,..., ,q

where and are defined with . The global condition (3.12) is equivalent to the strong discrete blockwise minimum principle

Hn zⁿ

0

:

q n

m m

m

g λ

=

=

∑

^g

(3.14) ⁿ( ,ⁿ_j ⁿ_j , ⁿ_j , ⁿ_j,1 , )_ijⁿ min ⁿ( ,ⁿ_j ⁿ_j , ⁿ_j , ⁿ_j,1 , ) ,

u U

H t y_θ y_θ z _−θ r dx H t_θ y_θ y_θ z _−θ u dx

Ω ∇ = ∈ Ω ∇

∫ ∫

1,...,

i= M , j=1,..., .N

4 Behavior in the limit

The following control approximation result is proved in [4].

Proposition 4.1 Under Assumptions 3.1 on Γ, for every r∈R, there exists a sequence (wⁿ∈Wⁿ) that converges to r inR.

Lemma 4.1 (Stability) Under Assumptions 2.2-3 and 3.1, if ∆t is sufficiently small, for every , we have the following inequalities, where the constants c are independent of and

rn∈Rⁿ n rⁿ (4.1) y_kⁿ ≤c, k =0,..., ,N

(4.2) ₁ ²

1

,

N

n n

j j

j

y y ₋ c

=

− ≤

∑

(4.3) ²

1 1

,

N

n n

j j

t y_θ

=

∆

∑

≤^c

(4.4) ²

0 1 N

n n

j j

t y

=

∆

∑

^≤c, (under the condition , for some constant C independent of , if

( )2

n n

t C h

∆ ≤ n θ =1/ 2),

(4.5) _{1 1}²

1 N n

j j

j

t y y ₋

=

∆

∑

− ^≤c

y , (with the condition ∆ ≤tⁿ C h( )^{n 2}).

Proof. Dropping the index for simplicity of notation, setting n v= ∆2θ t _j in the discrete equation, and using our assumptions on , we then have (if , then

, , ,0

a a b f b≠0

θ =1 and b y y( )_{j j} ≥0)

(4.6) θ( y_j −y_j−1 ²+ y_j ²− y_j−1 ²) ( (+ b t_jθ,y_jθ, ), ) r_j y_j

2 2

1 1

[ ( _j , _j ) ( , ) (1_{j j} ) ( _j , _j )]

t a y_θ y_θ θ a y y θ a y₋ y ₋

+∆ + − −

2θ t f t( ( ,_jθ y_jθ, ), ) 2r_j y_j θ t a0 _∞ y_jθ y_j

≤ ∆ + ∆ ∇

(1 _{j j} 1)

c t y_θ y_θ y

≤ ∆ + + _j

2 2

1

[1 _{j j} (1 1) ]

c t y_θ β y_θ y

≤ ∆ + + + +β _j ² hence, taking ²

2c

β ≤α , we get

(4.7) θ( y_j −y_j−1 ²+ y_j ²− y_j−1 ²)

2 2 2

2 1 1 1

[ ( , ) (1 ) (

2 ^{j j j j}

t α y_θ θ a y y θ a y ₋,y_j−)]

+∆ + − −

2 2 2

(1 _{j j} ) (1 _j 1 _j )

c t y_θ y c t y₋ y

≤ ∆ + + ≤ ∆ + + ²

2 2

1 1

(1 _{j j j} ),

c t y ₋ y y ₋

≤ ∆ + + −

and if in addition t 2

c

∆ ≤ θ

(4.8) 1 ₁ ^{2 2} ₁

( )

2 y^j y^j y^j y^j

θ − ₋ + − ₋ ²

2 2 2 2

2

1 1 1

[ 1 ( , ) (1 ) ( , )] (1

2 ^{j j j j j j}

t α y_θ θ a y y θ a y ₋ y ₋ c t y₋ ).

+∆ + − − ≤ ∆ +

By summation over j=1,...,k, we obtain, for θ >1/ 2

(4.9) ₁ ^{2 2 2 2} ₂

1 1

1 1

( 1 )

2 2

k k

j j k j j

j j

y y y α t y_θ tc y

θ ₋ α

= =

− + + ∆ + ∆

∑ ∑

²

1

'

k

∑

j=

2 2 2 2

0 1 0 1 1

1

(1 ) ^k (1 _j ),

j

y t y c t y

θ α θ ₋

=

≤ + ∆ − + ∆

∑

+ ^{with c' 0,>}

and for θ =1/ 2

(4.10) ₁ ^{2 2 2 2}

1 1 1

1 1

( )

2 2 2

k k

j j k j

j j

y y y α t y_θ

= − =

− + + ∆

∑ ∑

2 2 2

0 1 0 1 1

1

1 (1 ).

2 4

k

j j

y α t y c t y ₋

=

≤ + ∆ + ∆

∑

+

Since y_{0 1}, hence y₀ , remains bounded, using the discrete Bellman-Gronwall inequality (see 18]), we obtain inequality (i). The inequalities (4.2), (4.3), and (4.4) if θ >1/ 2, follow. By the inverse inequality (see [10]), the condition , and inequality (4.2), we get inequality (4.5)

( )2

tn C h

∆ ≤ ⁿ

(4.11) ₁ ² _{2 1} ²

1 1 1 1

.

N N N

j j j j j j

j j j

t y y t y y C y y c

− h −

= = =

∆

∑

− ≤ ∆

∑

− ≤

∑

− ^{−1 2} ≤

If θ =1/ 2, inequality (4.4) follows from inequalities (4.3) and (4.5).

For given values in a vector space, we define the piecewise constant and continuous piecewise linear functions

0,..., _N

v v

(4.12) v t₋( ) :=v_j−1, v t₊( ) :=v_j, v t_θ( ) : (1= −θ)v_j−1+θv_j, t∈I^on_j, j=1,..., ,N (4.13) _^( ) : ₁ ¹( _{j 1}),

n j

j n j

t t

v t v v v

t

−

− −

= + − −

∆ ^{, 1,..., .}

n

j j N

∈ =

t I

If b=0 (resp. b≠0), we suppose in the sequel that ∆ ≤tⁿ C (resp. ), with sufficiently small, so as to guarantee the results of Theorem 3.1 and Lemma 4.1.

( )2

n n

t C h

∆ ≤ C

Lemma 4.2 (Consistency of states and functionals) Under Assumptions 2.2-3 and 3.1, if rⁿ →r in R, then the corresponding discrete states y_^ⁿ,y₊ⁿ,y₊ⁿ,y₊ⁿ converge to

in (resp ) strongly if

yr 2( , ( ))4

L I L Ω L Q²( ) b≠0 (resp. b=0), in strongly, and

n

y_θ →yr L I V²( , ) (4.14) lim _mⁿ( )ⁿ _m( ),

n G r G r

→∞ = m=0,..., .q

Proof. By Lemma 4.1 (inequality (4.2) multiplied by ∆t), 0y₊ⁿ−y₋ⁿ → in strongly. Since, by inequality (4.3) in Lemma 3.3,

2( ) L Q y₋n and yⁿ₊ are bounded in , it follows that is also bounded in . By extracting subsequences,

we can suppose that and in weakly (hence in

weakly), for the same . The discrete state equation can be written in the form

2( , )

L I V y_^ⁿ L I V²( , )

^

yn → y y_θⁿ →y L I V²( , ) L Q²( ) y

(4.15) d ( ( ), ) (_^^{n n n}( ), )ⁿ ₁

y t v t v

dt = ψ , ∀ ∈vⁿ Vⁿ, a.e. in (0, ),T

in the scalar distribution sense, where the piecewise constant function ψⁿ is defined, using Riesz’s representation theorem, by

(4.16) (ψⁿ_j( ), ) :t vⁿ ₁ = −a y( ⁿ_jθ, ) ( ( )vⁿ − a t₀ ⁿ_jθ ^T∇yⁿ_jθ, ) ( ( ,vⁿ − b tⁿ_jθ yⁿ_jθ, ), )r_jⁿ vⁿ ( ( ,f tⁿ_jθ yⁿ_jθ, ), ),r_jⁿ vⁿ

+ in

o n

Ij , 1,..., .j= N