We propose now modifications of the method of extremal shift [Krasovskii, 1985], [Krasovskii, Subbotin, 1974, 1988], [A.N. Krasovskii, N.N. Krasovskii, 1995] in the direction of gen-eralized gradients of operators F, G, LA and prove optimality properties for designed trajectories. Let us note that rules of extremal aiming by quasigradients defined in the sense of Yosida-Moreau transformations were analyzed in the work [Garnysheva, Subbotin, 1994].
We give also the procedure realizing the extremal shift method in grid schemes. Values of generalized solutions of Hamilton-Jacobi equations, generalized gradients of local (linear, convex, concave) hulls and optimal feedbacks are calculated in parallel via the unique grid scheme.
In the grid scheme values of optimal feedbacks exist only at nodes. For constructing optimal trajectories which can slide between nodes of the grid it is necessary to solve the problem on interpolating values of optimal feedbacks to internal points. Different interpolants: piecewise constant, piecewise minimum, piecewise linear, are analyzed and their properties are studied. The question of correlation between spatial and temporal
grids is solved. In the general case the higher order density of the spatial mesh with respect to the temporal mesh provides optimality properties for designed trajectories.
The quasiconvexity property of grid functions approximating generalized solutions (value functions) provides the linear dependence of space-time grids.
Let us consider the problem of synthesizing optimal guaranteed feedbacks (t, x) → U0(t, x). For this purpose we use operatorGor operatorP M M∗which can be interpreted as minimax formulas defined on the local concave hull gor the local approximate function u respectively
G(t,∆, u)(x) =g(x) + min
y∈O(x,K∆)
min
s∈Dg(y){∆H(t, x, s) + g(y)−g(x)−< s, y−x >}= min
p∈P max
q∈Qg(y(t, x,∆, p, q)) (7.1) P M M∗ = min
p∈Pmax
q∈Q u(y(t, x,∆, p, q)) (7.2)
Here Euler spliney is determined by relations
y(t, x,∆, p, q) =x+ ∆(h(t, x) +B(t, x)p+C(t, x)q)
t∈T , t+ ∆∈T , t < ϑ, ∆>0, (t, x)∈Gr, r > K
Let us examine the approximation scheme with operator G (7.1) for partition Γ of interval T with step ∆
uΓ(ϑ, x) =σ(x), x∈Dϑ
uΓ(t, x) =G(t, ti+1−t, uΓ(ti+1,·))(x) (7.3) t∈[ti, ti+1), x∈Dt, i= 0, ..., N−1
Assume that the approximate function uΓ is constructed at all points (t, x), t ∈ Γ, x∈Dt. Let us define values of the optimal feedback U0=U0(t, x) at points (t, x) by the method of extremal shift using operatorGand its generalized gradients - superdifferentials of local concave hulls
U0=U0(t, x) =argmin
p∈P < s0, B(t, x)p > (7.4) s0 =s0(t, x, y0) =arg min
s∈Dg(y0){∆H(t, x, s) +g(y0)−< s, y0−x >} (7.5) y0 =y0(t, x) =arg min
y∈O(x,K∆)
min
s∈Dg(y){∆H(t, x, s) +g(y)−< s, y−x >} (7.6) Remark 7.1 According to the minimax formula (7.1) the optimal feedback U0 (7.4) is determined also by relations
U0 =U0(t, x) =argmin
p∈P max
q∈Qg(y(t, x,∆, p, q)) (7.7) Remark 7.2 The dual operators F, P M M∗ can be used in approximation schemes for constructing optimal feedbacks (t, x)→V0(t, x).
Lemma 7.1 For the optimal feedback U0 (7.4) the basic inequality is valid on the Euler spline y
maxq∈QuΓ(t+ ∆, y(t, x,∆, U0, q))≤uΓ(t, x) (7.8)
Proof.
For any control parameterq ∈Q we have according to definitions of operator G and feedback U0 the necessary chain of inequalities
uΓ(t+ ∆, y(t, x,∆, U0, q))≤g(y(t, x,∆, U0, q))≤max
q∈Qg(y(t, x,∆, U0, q)) =
sinf∈Rn(< s, x >+∆(< s, h(t, x)>+< s, B(t, x)U0 >+ maxq∈Q < s, C(t, x)q >)− inf
y∈Rn(< s, y >−g(y))) = min
y∈O(x,K∆)
min
s∈Dg(y)
(< s, x >+∆(< s, h(t, x)>+< s, B(t, x)U0 >+ maxq∈Q < s, C(t, x)q >)−< s, y >+g(y)) =
min
s∈Dg(y0)
(< s, x >+∆(< s, h(t, x)>+< s, B(t, x)U0 >+ maxq∈Q < s, C(t, x)q >)−< s, y0 >+g(y0)) =
∆(< s0, h(t, x)>+< s0, B(t, x)U0>+ max
q∈Q < s0, C(t, x)q >)−
< s0, y0−x >+g(y0) = ∆H(t, x, s0)−< s0, y0−x >+g(y0) =uΓ(t, x) 2 Inequality (7.8) can be transformed for a modification of the Euler spline y
y(t, x,∆, U0, q(·)) =x+ ∆(h(t, x) +B(t, x)U0) + Z t+∆
t
C(τ, x)q(τ)dτ
Here programming controlτ →q(τ) : [t, t+∆)→Qis a Lebesgue measurable function.
Lemma 7.2 The optimal feedback U0 (7.4) provides estimates
supq(·)uΓ(t+ ∆, y(t, x,∆, U0, q(·)))≤uΓ(t, x) +LwL2K∆2 (7.9)
Proof.
Taking into account Lipschitz conditions (H2), (H4) for the HamiltonianH, convexity of the control set Qand using the integral mean value theorem we obtain relations
Z t+∆
t
C(τ, x)q(τ)dτ= Z t+∆
t
(C(τ, x)−C(t, x))q(τ)dτ+ C(t, x)
Z t+∆
t
q(τ)dτ=ε(∆)∆ +C(t, x)∆q, q∈Q kε(∆)k ≤L2K∆
Then the basic inequality (7.8) and the Lipschitz continuity of the approximate function uΓ with the constant Lw imply the necessary inequality (7.9). 2
We apply now basic inequalities (7.8), (7.9) to derivation of optimality properties for Euler trajectories generated by feedback U0
x(·) ={x(t, t∗, x∗, U0, q(·)) : t∈Γ\[t∗, ϑ], t∗ ∈Γ, x∗∈Dt∗} (7.10) x(ti+1) =x(ti+ ∆) =x(ti) + ∆(h(ti, x(ti)) +B(ti, x(ti))U0) +
Z ti+∆
ti
C(τ, x(ti))q(τ)dτ, x(t∗) =x∗, ti, ti+1∈Γ
Theorem 7.1 For any partitionΓ, initial position (t∗, x∗) and Lebesgue measurable con-trol τ → q(τ) : [t∗, ϑ) → Q the trajectory (7.10) generated by feedback U0 satisfies the estimate
σ(x(ϑ))≤uΓ(t∗, x∗) +LwL2K(ϑ−t∗)∆ (7.11) and, hence, by Theorem 3.2
σ(x(ϑ))≤w(t∗, x∗) +C∆1/2+LwL2K(ϑ−t∗)∆ (7.12) Fixing an arbitrary number ε >0one can indicate step ∆of partition Γ providing the estimate
σ(x(ϑ))< w(t∗, x∗) +ε
Proof.
By induction inequality (7.9) implies the estimate
uΓ(ti, x(ti, t∗, x∗, U∗, q(·)))≤uΓ(t∗, x∗) + (ti−t∗)∆−1LwL2K∆2 which provides relation (7.11) when ti =ϑ.
Inequality (7.12) follows from estimates (3.13), (7.11). 2 Remark 7.3 For trajectories x(·) generated by feedback UP
UP =UP(t, x) =argmin
p∈Pmax
q∈QuΓ(t+ ∆, y(t, x,∆, p, q)) (7.13) in the approximation scheme with minimax operator P M M∗ (7.2) estimates (7.9), (7.12) are fulfilled.
In reality approximation scheme (7.3) can be realized only at nodes of gridGR(t) but not in the whole domain Dt,t∈ Γ. Let us assume that gridGR(t), t∈Γ is rectangular and uniform
GR(t) ={y∈Dt: y=X(m1e1+...+mnen)γ∆} (7.14) mi= 0,±1,±2, ..., i= 1, ..., n
ei = (e1i, ..., eni), eii= 1, eji = 0, i6=j
We define values of operator G only at nodes yj of grid GR(t) and interpolate them linearly into domain Dt∗
D∗t ={y∈Rn: y= Xn j=0
αjyj, yj ∈GR(t), αj ≥0, Xn j=0
αj = 1} according to the given simplicial partition Ω.
Practically we introduce operator y→G∗(t,∆, u)(y) :Dt∗→R G∗(t,∆, u)(y) =
Xn j=0
αjG(t,∆, u)(yj) (7.15)
y∈Dt∗, y = Xn j=0
αjyj, yj ∈GR(t), αj ≥0, Xn j=0
αj = 1 Here numbers αj =αj(Ω) and nodes yj =yj(Ω) are determined by partition Ω.
Let us consider the approximation scheme with operator G∗ (7.15) for partition Γ of interval T with step ∆
u∗Γ(ϑ, y) =σ∗(y) = Xn j=0
αjσ(yj), y ∈D∗ϑ, y = Xn j=0
αjyj Xn
j=0
αj = 1, αj =αj(Ω)≥0, yj =yj(Ω)∈GR(ϑ), j= 0, ..., n
u∗Γ(t, x) =G∗(t, ti+1−t, u∗Γ(ti+1,·))(x) (7.16) t∈[ti, ti+1), x∈Dt∗, i= 0, ..., N−1
Assume that the approximate function u∗Γ is calculated at all points (t, y), y ∈ Dt∗, t ∈Γ. Let us determine at first values of the optimal feedback U∗ =U∗(t, x) at nodes x of grid GR(t),t∈Γ using operator G(t,∆, u∗Γ(t+ ∆,·))(x) =G∗(t,∆, u∗Γ(t+ ∆,·))(x)
U∗ =U∗(t, x) =argmin
p∈P < s∗, B(t, x)p > (7.17)
s∗ =s∗(t, x) =arg min
s∈Dg(y∗){∆H(t, x, s) +g(y∗)−< s, y∗−x >}
y∗ =y∗(t, x) =arg min
y∈O(x,K∆)
min
s∈Dg(y){∆H(t, x, s) +g(y)−< s, y−x >}
We define the optimal feedback UC(t, y) in domain y ∈ D∗t, t ∈ Γ by the piecewise constant interpolation of values {U∗(t, x)} (7.17) calculated at nodes x ∈ GR(t), t ∈ Γ neighboring to points y
UC(t, y) =U∗(t, x), x=x(y) =arg min
z∈GR(t)ky−zk (7.18) Let us introduce the Euler trajectory
y(·) ={y(t, t∗, y∗, UC, q(·)) : t∈Γ\[t∗, ϑ]} (7.19) generated by feedback UC (7.17), (7.18) and an arbitrary Lebesgue measurable control τ →q(τ) : [t∗, ϑ)→Q
y(ti+1) =y(ti+ ∆) =y(ti) + ∆(h(ti, y(ti)) +B(ti, y(ti))UC) + Z ti+∆
ti
C(τ, y(ti))q(τ)dτ, y(t∗) =y∗, ti, ti+1∈Γ\[t∗, ϑ] (7.20) For trajectoryy(·) (7.19) we define accompanying points
(x−(ti), x+(ti+1)) (7.21)
by relations
x−(ti) =x(y(ti)) =arg min
z∈GR(t)ky(ti)−zk (7.22) x+(ti+1) =x−(ti) + ∆(h(ti, x−(ti)) +B(ti, x−(ti))UC) +
Z ti+∆
ti
C(τ, x−(ti))q(τ)dτ, ti, ti+1∈Γ\[t∗, ϑ] (7.23) Lemma 7.3 For trajectoryy(·)(7.19) and accompanying points{(x−(ti), x+(ti+1))}(7.21) the estimate
ky(ti+1)−x+(ti+1)k ≤(1 +L1∆)ky(ti)−x−(ti)k+ 2L2K∆2 (7.24) is valid.
Lemma 7.4 Let parameter γ of grid GR(t), t∈Γ be an infinitesimal value with respect to step ∆
γ =ε(∆), lim
∆→0ε(∆) = 0 for example,
γ =ρ∆a, a >0, ρ >0 (7.25)
and, hence, step h of the spatial gridGR(t) is a high order infinitesimal value
h=ρ∆1+a (7.26)
with respect to the time step ∆.
Then the basic inequality for functionu∗Γ is valid u∗Γ(ti, y(ti))≥u∗Γ(ti+1, y(ti+1))−Lw(
√n
2 (2 +L1∆)ρ∆a+ 3L2K∆)∆ (7.27)
Proof.
The Lipschitz continuity of function u∗Γ and relations (7.9), (7.24) imply inequalities u∗Γ(ti, y(ti))≥u∗Γ(ti, x−(ti))−Lwky(ti)−x−(ti)k ≥
u∗Γ(ti+1, x+(ti+1))−LwL2K∆2−Lwky(ti)−x−(ti)k ≥ u∗Γ(ti+1, y(ti+1))−Lw(2 +L1∆)ky(ti)−x−(ti)k −3L2K∆2 Relation (7.25) provides the inequality
ky(ti)−x−(ti)k ≤
√n 2 ρ∆1+a for trajectoryy(·) and accompanying pointsx−(ti).
The last two inequalities give estimate (7.27). 2
Using estimate (7.27) one can prove the following proposition for trajectoryy(·) (7.19) generated by the optimal feedback UC (7.17), (7.18).
Theorem 7.2 For any partitionΓ, gridGR(t),t∈Γwith high order infinitesimal param-eters (7.26), initial position(t∗, y∗)and Lebesgue measurable controlτ →q(τ) : [t∗, ϑ)→Q the trajectory y(·) (7.19) generated by the optimal feedback UC (7.17), (7.18) with the piecewise constant interpolation satisfies the estimate
σ(y(ϑ))≤u∗Γ(t∗, y∗) +ϕ(∆) (7.28)
ϕ(∆) = (ϑ−t∗)Lw(
√n
2 (2 +L1∆)ρ∆a+ 3L2K∆), lim
∆→0ϕ(∆) = 0 and, hence, by Theorem 3.4
σ(y(ϑ))≤w(t∗, y∗) +C∗∆1/2+ϕ(∆) (7.29) Fixing an arbitrary number ε > 0 one can indicate step ∆ of partition Γ which provides the estimate
σ(y(ϑ))< w(t∗, y∗) +ε
Proof.
By induction inequality (7.27) implies the estimate
u∗Γ(ti, y(ti, t∗, y∗, UC, q(·)))≤u∗Γ(t∗, y∗) + (ti−t∗)∆−1Lw(
√n
2 (2 +L1∆)ρ∆a+ 3L2K∆)∆
which provides relation (7.28) when ti =ϑ.
Inequality (7.29) follows directly from relations (3.20), (7.28). 2
Remark 7.4 One can use other piecewise constant interpolations of feedbackU∗ =U∗(t, x) (7.17). For example, the optimal feedbackUE(t, y)can be interpolated into domain y∈D∗t, t ∈Γ by controls U∗(t, x) calculated at nodes x ∈GR(t) with least values of the approxi-mate function u∗Γ in the neighboring simplex
UE(t, y) =U∗(t, x), x=x(y) =argmin
yj
u∗Γ(yj) (7.30)
y= Xn j=0
αjyj, yj =yj(Ω)∈GR(t), Xn j=0
αj = 1, αj =αj(Ω)≥0 In this case estimates (7.27) and (7.29) can be rewritten in the following way
u∗Γ(ti, y(ti))≥u∗Γ(ti+1, y(ti+1))−Lw(
√n
2 (1 +L1∆)ρ∆a+ 3L2K∆)∆ (7.31) σ(y(ϑ))≤w(t∗, y∗) +C∗∆1/2+ϕ∗(∆) (7.32) ϕ∗(∆) = (ϑ−t∗)Lw(
√n
2 (1 +L1∆)ρ∆a+ 3L2K∆)
Remark 7.5 For trajectory y(·) generated by feedback UP (7.33) which is determined at nodes x of grid GR(t) by minimax operator P M M∗ (7.2)
UP =UP(t, x) =argmin
p∈Pmax
q∈Qu∗Γ(t+ ∆, y(t, x,∆, p, q)) (7.33) and interpolated into domain Dt∗ by piecewise constant rules (7.18) or (7.30) one can obtain estimates (7.27), (7.29) or (7.31), (7.32).
Let us consider the case of quasiconvex approximate functions y→u∗Γ(t, y) :Dt∗→R, t∈Γ for gridsGR(t) with the linear dependence of space-time stepsh, ∆
h=γ∆ (7.34)
Here γ is a fixed constant.
Let us formulate the property of quasiconvexity for functions y→u∗Γ(t, y).
Conjecture 7.1 Approximate functions y →u∗Γ(t, y) :Dt∗→R,t∈Γ satisfy the convex-ity condition up to the infinitesimal value µ∆1+b,b >0, µ >0 in domains with radius ν∆
- the quasiconvexity condition Xn j=0
αju∗Γ(t, zj) +µ∆1+b ≥u∗Γ(t, Xn j=0
αjzj) (7.35)
Xn j=0
αj = 1, αj ≥0, j = 0, ..., n
kzk−zlk ≤ν∆, zk, zl∈Dt∗, k, l= 0, ..., n
ν >2K+√ Let us introduce the Euler trajectory
z(·) ={z(t, t∗, y∗, UL, q(·)) : t∈Γ\[t∗, ϑ]} (7.38) generated by feedbackUL(7.37) and an arbitrary Lebesgue measurable controlτ →q(τ) : [t∗, ϑ)→Q
z(ti+1) =z(ti+ ∆) =z(ti) + ∆(h(ti, z(ti)) +B(ti, z(ti))UL) + Z ti+∆
ti
C(τ, z(ti))q(τ)dτ, z(t∗) =z∗, ti, ti+1∈Γ\[t∗, ϑ] (7.39) Lemma 7.5 The basic estimate
u∗Γ(ti, z(ti))≥u∗Γ(ti+1, z(ti+1))−µ∆1+b−Lw(√
nL1γ + 3L2K)∆2 (7.40) is valid for trajectory z(·) (7.38), (7.39).
Proof.
According to the property of quasiconvexity (7.35), the Lipschitz continuity of function u∗Γ and definition of feedback U∗ (7.17) we obtain the necessary chain of inequalities
u∗Γ(ti+1, z(ti+1)) =u∗Γ(ti+ ∆, z(ti) + ∆(h(ti, z(ti)) +B(ti, z(ti))UL) +
and pointszj are disposed in the ball of radius ν∆
kzk−zlk ≤(1 +L1∆)kyk−ylk+ 2L2K∆2+
∆kB(ti, yk)Uk∗k+ ∆kB(ti, yl)Ul∗k< ν∆, k, l= 0, ..., n 2 The basic estimate (7.40) implies the following proposition.
Theorem 7.3 Assume that Conjecture 7.1 is fulfilled for approximate functions u∗Γ in schemes with operator G∗ (7.15) and the linear dependence (7.34) of space-time steps.
Then for any initial position (t∗, y∗) and Lebesgue measurable control τ → q(τ) : [t∗, ϑ)→Q the trajectory z(·) (7.38) generated by the optimal feedback UL (7.17), (7.37) with the linear interpolation satisfies the estimate
σ(z(ϑ))≤u∗Γ(t∗, z∗) +ψ(∆) (7.41)
ψ(∆) = (ϑ−t∗)(µ∆b+Lw(√
nL1γ+ 3L2K)∆), lim
∆→0ψ(∆) = 0 and, hence, by Theorem 3.4
σ(z(ϑ))≤w(t∗, z∗) +C∗∆1/2+ψ(∆) (7.42) Fixing an arbitrary number ε > 0 one can indicate step ∆ of partition Γ which provides the estimate
σ(z(ϑ))< w(t∗, z∗) +ε
Proof.
By induction inequality (7.40) implies relations
u∗Γ(ti, z(ti, t∗, z∗, UL, q(·)))≤u∗Γ(t∗, z∗) + (ti−t∗)∆−1(µ∆1+b+Lw(√
nL1γ+ 3L2K)∆2) which leads to the estimate (7.41) when ti =ϑ.
Inequality (7.42) follows directly from relations (3.20), (7.41). 2
Remark 7.6 Let Conjecture 7.1 be fulfilled for approximate functionsu∗Γin schemes with operator P M M∗ (7.2). Then trajectories z(·) generated by feedback UP (7.33) which is defined at nodes x of grid GR(t) by minimax operator P M M∗ (7.2) and interpolated linearly (7.37) into domain Dt∗ satisfy estimates (7.40), (7.42).
Remark 7.7 Assume that Conjecture 7.1 is true for approximate functionsu∗Γin schemes with operator LA (6.29). Consider trajectoriesz(·)generated by feedbackUA (7.43) which is defined at nodes x of grid GR(t) by operator LA(6.29) - the minimax formula on local linear hulls
UA=UA(t, x) =argmin
p∈P max
q∈QL(y(t, x,∆, p, q)) =argmin
p∈P < A, B(t, x)p > (7.43) and interpolated linearly (7.37) into domain D∗t. Then estimates (7.40), (7.42) are valid.
Proof.
Conditions of quasiconvexity (7.35), (7.36) imply inequalities L(y)≥u∗Γ(t, y)−µ∆1+b, L(y)≤g(y) y∈O(x, K∆), x∈GR(t), t∈Γ and, therefore, provide estimates (7.40), (7.42). 2
Conclusion
The unique grid scheme for constructing value functions and control synthesis is proposed for solving optimal guaranteed control problems which arise in mechanics, mathematical economics, evolutionary biology. Finite difference operators based on constructions of non-smooth analysis - subdifferentials of local convex hulls, superdifferentials of local concave hulls and their modifications, are elaborated for local approximations of nondifferentiable value functions and its generalized gradients (dual vectors). Control synthesis is obtained by the method of extremal shift in the direction of generalized gradients. Convergence of approximation schemes is proved by using methods of the theory of Hamilton-Jacobi equations. Properties of space-time grids providing optimality of designed trajectories are examined.
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