Set-Membership Method for Discrete Optimal Control

Set-Membership Method for Optimal Control

R´emy Guyonneau, S´ebastien Lagrange, Laurent Hardouin, Mehdi Lhommeau

Laboratoire d’Ing´enierie des Syst`emes Automatis´es (LISA), Universit´e d’Angers, 62 avenue Notre Dame du Lac, Angers,France

{remy.guyonneau, sebastien.lagrange, laurent.hardouin, mehdi.lhommeau}@univ-angers.fr

Keywords: Non-Linear Systems, Interval Analysis, Guaranteed Numerical Integration, Optimal control

Abstract: The objective of this paper is twofold: to propose a new approach for computingCt0,tf the subset of initial states of a system from which there exists at least one trajectory reaching a targetTin a finite timet_f from a timet0, and, using that work and given a cost function, to estimate an enclosure of the optimal control vector from an initial state ofCt₀,t_f to the target. Whereas classical methods do not provide any guaranteed on the set of state vectors that belong to theCt0,t_f, interval analysis and guarantee numerical integration allow us to avoid any indetermination. We present an algorithm able to provide guaranteed characterizations of the innerC⁻_t0,tf and an the outerC⁺_t0,tf ofCt₀,t_f, such thatC⁻_t0,tf ⊆Ct₀,t_f⊆C_t⁺_0,tf. In addition to that, the presented algorithm is extended in order enclose the optimal control vector of the system, form an initial state to the target, by a set of trajectories.

1 INTRODUCTION

We consider a control system, defined by the differ- ential equation

˙

x(t) =f(x(t),u(t)) (1) wherex(t)∈Rⁿbe the state vector of the system and u(t)∈Ube the control vector. This system is stud- ied over a bounded timet∈[t₀,t_f]. Associated to the differential equation (1) we define theflow map

ϕ(t₀,t;x₀,u) =x(t), (2) wherex(t)denotes the solution to (1) with the initial conditionx(t₀) =x₀and the control functionu∈U, whereU⁼{u:[t₀,t]→U|t≥t₀anduis piecewise continuous} denotes the set of admissible controls.

GivenX₀a set of possible initial valuesx₀, the reach- able set of the system (1) at the timetis

ϕ(t₀,t;X₀,U) ={ ϕ(t₀,t;x₀,u)|ϕ(t₀,t₀;x₀,u) =x₀ andϕ:[t₀,t]×X₀×U→Rⁿ is a solution of (1) for some u∈U}.

(3) The trajectory fromttotis defined by

ϕ([t,t];X,U) ={ X˜ ⊆Rⁿ|

∃t∈[t,t],X˜ =ϕ(t,t;X,U)}. (4)

LetK⊂Rⁿbe a state constraintx(t)∈K, andTbe a compact set inK(the target).C_t0,tf corresponds to the subset of initial states ofKfrom which there exists at least one solution of (1) reaching the targetTin finite timetf from a timet0:

C_t0,tf ={x₀∈K|∃u∈U^,ϕ(t₀,t_f;x₀,u)∈T} (5) The first objective of this paper is to compute an inner and outer approximations,C⁻_t

0,t_f andC_t⁺

0,t_f, ofC_t0,tf, as done in (Lhommeau et al., 2011; Delanoue et al., 2009). Such problems of dynamics control under con- straints refer to viability theory (Aubin, 2006) (see (Aubin, 1990) for a survey). The proposed method to characterizeC_t0,tf has the advantage that it is guaran- teed whereas numerical methods give only an approx- imation. An interval analysis based method is used to compute the approximations, such that C⁻_t0,tf ⊆ C_t0,t_f ⊆C⁺_t0,tf, by using guaranteed numerical integra- tion (VNODE-LP¹).

In a second part, we got interested to the optimal con- trol problem. Given a cost functionJ and an initial statex₀∈C_t0,tf, we propose a numerical method to evaluate an enclosure of the optimal controlu∈U such thatϕ(t₀,t_f;x₀,u)∈T.

The paper is organised as follow. First some interval analysis tools are presented in Section 2 as they are

1A C++ package for computing bounds on solutions in Initial Value Problems for Ordinary Differential Equations, by N. Nedialkov.

used to compute the inner and outer approximations.

Section 3 presents the proposed algorithm to compute C_t0,tf and is followed by experimental results in Sec- tion 4. Finally Section 5 discusses about the optimal control problem and Section 6 concludes this paper.

2 INTERVAL ANALYSIS

Interval analysis for ordinary differential equations was introduced by Moore (Moore, 1966) (See (Nedi- alkov et al., 1999) for a description and bibliography on this topic). These methods provide numerically re- liable enclosures of the exact solution of differential equations.

Interval analysis usually considers only closed inter- vals. The set of these intervals is denotedIR. An in- terval is usually denoted using brackets. An element of an interval[x]is denoted byx. An interval vector (box)[x]ofRⁿis a Cartesian product ofnintervals. If [x] = [x₁,x₁]× ··· ×[x_n,x_n]is a box, then its width is

w([x]) =w([x₁])× ··· ×w([x_n]), (6) wherew([x_i]) =x_i−x_i. The set of all boxes ofRⁿis denoted byIRⁿ.

TheBisect()function divides an interval[x]into two intervals[x₁]and[x₂]such as[x₁]∪[x₂] = [x],[x₁]∩ [x₂] =/0andw([x₁]) =w([x₂]).

The main concept of interval analysis is the extension of real functions to intervals, which is defined as fol- lows. Letf:Rⁿ→R^mbe a continuous real function, and[f]:IRⁿ→IR^m be an inclusion function. Then [f]is an inclusion function offif and only if for every [x]∈IRⁿ,{f(x)|x∈[x]} ⊆[f]([x]).

Hence, an interval inclusion allows computing en- closures of the image of boxes by real functions. It now remains to show how to compute such inclu- sions. The first step is to compute formally the in- terval extension of elementary functions. For exam- ple, we define[x,x] + [y,y]:= [x+y,x+y]. Similar simple expressions are obtained for other functions like −,×,÷,xⁿ,√

x,exp,··· This process gives rise to the so-calledinterval arithmetic(see (Jaulin et al., 2001)).

Then, an interval inclusion for real functions com- pound of these elementary operations is simply ob- tained by changing the real operations to their inter- val counterparts. This interval inclusion is called the natural extension.

Given a bounded setEof complex shape, one usu- ally defines an axis-aligned box or paving, i.e. an union of non-overlapping boxes,E⁺which contains the setE: this is known as the outer approximation of it. Likewise, one also defines an inner approximation

E⁻ which is contained in the setE. Hence we have the following property

E⁻⊆E⊆E⁺ (7)

3 CHARACTERIZATION OF C

The proposed algorithm considers a discretization of the time

t_k=t₀+k×δt,t_k≤t_f,k∈ {1,···,m}, (8) and for each time’s stept_kcomputes a gridding ofK (aslice), notedS(t_k). The resolution of the gridding is δK= (δx₁,···,δx_i,···,δxn)whereδx_i corresponds to the resolution of thei^thdimension ofK(Figure 1).

A cells_iofS(t_k)can be

- unreachable if no statex in this cell allows the system to reach the target at timet_f, for all possi- ble input vectors. The set of all theunreachable cells ofS(t_k)is notedS^u(t_k)

S^u(t_k) ={s_i∈S(t_k)|∀u∈U,ϕ([t_k,t_f];s_i,u)∩T=/0}

(9) - reachableif for all the statesxof this cell it exists an input vector that allows the system to reach the target at timet_fwith a trajectory entirely included in the state space domainK. The set of all the reachablecells ofS(t_k)is notedS^r(t_k).

S^r(t_k) = {s_i∈S(t_k)|∃u∈U, ϕ(t_k,t_f;s_i,u)⊆Tand

ϕ([t_k,t_f];s_i,u)⊆K} (10) - indeterminateif it is neitherreachableorunreach- able. The set of all theindeterminatecells ofS(t_k) is notedSⁱ(t_k).

Sⁱ(t_k) =S(t_k)\(S^r(t_k)∪S^u(t_k)) (11) It can be noticed that

C⁻_t0,tf =S^r(t₀),

C⁺_t0,tf =S^r(t₀)∪Sⁱ(t₀). (12) The time’s stepδt of the input vector (step time dur- ing which one the input vector is continuous and bounded) and the resolutionsδxi,i=1,···,n, are de- fined by the desired precision of theC⁺_t

0,t_f andC⁻_t

0,t_f

characterizations. Note thatC⁻_t0,tf can be empty if the precision is too rough.

Figure 1: An example of slice set and sliceS(tk), with K= ([x1,x1],[x2,x2]) a two dimensional state space do- main. The following color scheme is held for all the figures of this paper: blue (dark grey)→unreachable, red (medium grey)→reachable, yellow (light grey)→indeterminate.

3.1 The C

Algorithm

We propose an iterative algorithm (Algorithm 1) that computes for each slice S(t_k), three subsets S^r(t_k), S^u(t_k)andSⁱ(t_k)such as

S(t_k) =S^r(t_k)∪S^u(t_k)∪Sⁱ(t_k) S^r(t_k) ={s_i⊆S(t_k)|s_iisreachable} S^u(t_k) ={s_i⊆S(t_k)|s_iisunreachable} Sⁱ(t_k) ={s_i⊆S(t_k)|s_iisindeterminate}

(13)

Those subsets are computed fromt_f tot₀using guar- antee numerical integration. After the initialization of the subsets at timet_f the other subsets at timet_kare built using the reachability information of the cells s_i ⊆S(t_k+1). Note that the ADD algorithm is pre- sented in Section 3.2.

Lines 1 to 3 of the Algorithm 1 initialise the three subsets of the sliceS(t_f). For this particular slice, the reachablecells are the ones included in the target, the unreachablecells are the ones that do not intersect the target and theindeterminatecells are all the oth- ers ones (the cells intersecting the target without been included). Then lines 4 to 24, the others slice subsets are built. Line 6 it can be notices that all the possible control vectors are considered to determine the reach- ability of the cells. Line 8 to 24 aSet Inversion Via Interval Analysis approach (Jaulin and Walter, 1993) is used to determinate the reachability of the current slice cells. It can be noticed that the computation ofϕ line 10 is done using guaranteed numerical integration (VNODE-LP). Line 14, a cell can be reachable only if the trajectory is included in the state spaceK. Usu- ally the computation of the inclusion flow is based on the Banach fixed-point theorem and the application of the Picard-Lindelof operator (see (Berz and Makino, 1998; Nedialkov et al., 1999) for details). Line 21, the box is bisected among the grid. It means that, line 20, the box[x]can not be bisected if it contains only one cell.

Algorithm 1:COMPUTATION OFC_t0,tf Data:K,T,t₀,t_f,U

1 S^r(t_f) ={s_i∈S(t_f)|s_i⊆T}; 2 S^u(t_f) ={s_i∈S(t_f)|s_i∩T=/0}; 3 Sⁱ(t_f) =S(t_f)\(S^r(t_f)∪S^u(t_f));

4 fort_k←t_f−1tot₀do 5 L⁼/0;

6 forall the[u]∈U do

7 L^.add(K);

8 whileLis not emptydo

9 [x] =L.pop out();

10 [x]^∗=ϕ(t_k,t_k+1;[x],[u]);

11 if[x]^∗∩K=/0then

12 ∀si⊆[x],ADD(S^u(t_k),si);

13 else if[x]^∗⊆S^r(t_k+1)then 14 ifϕ([t_k,t_k+1];[x],[u])⊆K then 15 ∀s_i⊆[x],ADD(S^r(t_k),s_i);

16 else

17 ∀s_i⊆[x],ADD(Sⁱ(t_k),s_i);

18 else if[x]^∗⊆S^u(t_k+1)then 19 ∀s_i⊆[x],ADD(S^u(t_k),s_i));

20 else if[x]can be bisectedthen 21 ([x₁],[x₂]) =BISECT([x]);

22 L^.add([x1]),L^.add([x2]);

23 else

24 ∀s_i⊆[x],ADD(Sⁱ(t_k),s_i));

Result:{S(t_k)},t_k=t₀,···,t_f.

The result of the algorithm has to be interpreted as C⁺_t0,tf =S^r(t₀)∪Sⁱ(t₀)

C⁻_t0,tf =S^r(t₀) (14) The Figure 2 represents three cases of the Algo- rithm 1:

- [x₁]^∗⊂S^u(t_k+1), then{s_i⊆S(t_k)|s_i⊆[x₁]}can be added toS^u(t_k)(Line 19 of the algorithm), - [x₂]^∗⊂S^r(t_k+1), then{s_i⊆S(t_k)|s_i⊆[x₂]}can be

added toS^r(t_k) [x₂]^∗(Line 15 of the algorithm), - [x₃]^∗ is neither included inS^u(t_k+1)or S^r(t_k+1),

and the box[x₃]is too small to be bisected, thus {s_i∈S(t_k)|s_i⊆[x₃]}can be added toSⁱ(t_k)(Line 24 of the algorithm).

3.2 The ADD algorithm

The slide’s cell reachability is updated regards to all the possible control vectors[u]∈U. For a given cell,

Figure 2: The reachability of the cells si⊆S(t_k+1) are used to build the subsets of the sliceS(t_k). Denote that [xi]^∗=ϕ(t_k,t_k+1;[xi],[u]),i=1,2,3, with[x₁]^∗⊂S^u(t_k+1), [x2]^∗⊂S^r(t_k+1)and[x3]^∗is neither included inS^u(t_k+1)or S^r(t_k+1).

the reachability information can be different consid- ering two different control vectors. That is why it is needed to considerprioritiesfor the update of the reachability information of a cell and thus the com- puting of the three subsets S^r(t_k),S^u(t_k) and Sⁱ(t_k).

That is the purpose of the ADD function. For example if a control vector[u₁] leads to areachabilityinfor- mation for a cells_i⊆S(t_k)whereas a control vector [u₂]leads to aindeterminateinformation for the same cell, this cells_i belongs toS^r(t_k)(is reachable) be- cause it has been proved that it exists a control vector, [u₂], that leads the cell toS^r(t_k+1). Figure 3 presents the several reachability information priorities.

Figure 3: The reachability information priorities. A cell notedunreachablecan be updated toreachableorindeter- minate, a cell notedindeterminatecan only be updated to reachableand a cell notedreachablecan not be updated at all.

4 EXPERIMENTATION

In order to validate the proposed method the algo- rithm has been implemented in C++ using VNODE- LP library. Be considered the following two- dimensional system

x˙₁ = x₂+v×cos(θ),

˙

x2 = sin(x₁) +v×sin(θ), (15) the following input vector

slice S(9.5) S(9) S(8.5) S(8) time (s) 2.56 5.55 11.95 22.6 slice S(7.5) S(7) S(6.5) S(6) time (s) 40.66 61.78 71.86 75.6 slice S(5.5) S(5) S(4.5) S(4) time (s) 76.41 79.6 82.88 87.89

slice S(3.5) S(3) S(2.5) S(2) time (s) 93.96 98.59 102.79 105.87

slice S(1.5) S(1) S(0.5) S(0) time (s) 106.78 108.5 109.23 111.73

Table 1: Slices computation time











U={[u₁],[u₂],[u₃]} [u_i] = ([v_i],θi),

[u₁] = ([−0.25,0.25],π/2), [u₂] = ([3.75,4.25],π/2), [u₃] = ([9.75,10.25],π/2),

(16)

the following parameters







δt=0.5, t0=0,tf =10, δx1=0.5, δx2=0.5, K= ([−30,30],[−30,30]),

(17)

and the following target

T= ([−15.1,−9.9],[1.9,7.1]). (18) The Figure 4 shows four slices. The PC we use has two processors (Intel(R) Core(TM)2 CPU 6420 @ 2.13 Ghz), and it takes 1457s (24min 17s) to compute all the slices S(t_k). The details of the computation time are presented in the Table 1.

The sliceS(0)provides theC⁺_t0,tf =S^r(0)∪Sⁱ(0)and C⁻_t

0,t_f =S^r(0)characterizations ofC_t0,tf. Note that it is possible to increase the precision of the approxima- tion ofC_t0,tf by reducing the values ofδx₁ andδx₂.

Figure 4: Four slices: (from left top to right bottom)S(9.5), S(7.5),S(5)andS(2.5).

5 OPTIMAL CONTROL ENCLOSURE

The previous algorithm computes two guaranteed ap- proximationsC⁻_t0,tf andC⁺_t0,tf ofCt₀,t_f. It is possible to extend this algorithm to be able to deal with optimal control. Considering a given cost functionJ(x,u), the idea of the algorithm is to enclose the cost of the tra- jectories that allow to reach a cells_i⊆S(t_k+1)from a cells_j ⊆S(t_k). To simplify we assume that the cost function to minimize is

J= Z

u²dt. (19)

It can obviously be extended to other cost functions.

Instead of characterizing the cells with the reachabil- ity of the target this section provides a method to add the input control that could be used to reach the target.

The modifications of the previous algorithm are pre- sented in Subsection 5.1, Subsection 5.2 details how to build a graph using the added control vectors, and Subsection 5.3 explains how to use the graph to en- close the optimal control input.

5.1 The Algorithm Modifications

The idea is to use theCt₀,t_f computation to enclose the optimal trajectory from an initial state[x₀]∈Ct₀,t_f

to the targetT. To this end it is needed to slightly modify the presented algorithm. The purpose of this new algorithm is to define for all the cellss_i⊆S(t_k)a set of input vectorsU(s_i)that leads the cell toS^r(t_k+1) orSⁱ(t_k+1)(Figure 5):

U(s_i) ={[u]∈U|ϕ(t_k,t_k+1;s_i,[u])6⊆S^u(t_k+1)}, (20) withs_i∈S(t_k),t_k<t_f.

Each time a cells_i⊆S(t_k)may be added toSⁱ(t_k)or S^r(t_k)(lines 15,17 and 24 of the Algorithm 1) the cur- rent input vector[u]has to be added toU(s_i).

Figure 5: Example of added control vectors for a cells_i⊆ S(t_k):U(s_i) ={[u₂],[u₃]},[u₁]is not relevant since it leads toS^u(t_k+1). Note that[sj]^∗=ϕ(t_k,t_k+1;si,[uj]),j=1,2,3.

Considering the cost function it is possible to asso- ciate a costJ([u])to each control vector[u]∈U(s_i),

s_i⊆S(t_k). In the following a control vector[u]will be abusively associated to its costJ([u]).

5.2 A Graph Building

Given an initial state[x₀]∈C_t0,tf, the idea is to build a graph starting with a noden₀(t₀)and ending with a noden_T(t_f)(Figure 7), such as

n₀(t₀) ={s_i⊆S(t₀)|s_i∩[x₀]6=/0}

n_T(t_f) ={s_i⊆S(t_f)|s_i∩T6=/0} (21) N defines the set of nodes of the graph. A node n_i(t_k)∈Nis defined by a set of cellss_i⊆S(t_k). Two nodesni(t_k)andnj(t_k+1)are linked if

∃[u]∈U|∀s_i∈n_i(t_k),ϕ(t_k,t_k+1;s_i,[u])⊆n_j(t_k+1) (22) withJ([u])the weight of the edge that links the two nodesni(t_k)andnj(t_k+1).

It is possible to define a set of control vectorU(n_i(t_k)) for a noden_i(t_k)∈Ncorresponding to all the control vectorsU(s_i)of all the cellss_i⊆n_i(t_k):

U(n_i(t_k)) ={U(s_i),∀si∈ni(t_k)}. (23) Note that for efficiency reason it is recommended to avoid control vector redundancy in U(n_i(t_k)),

∀n_i(t_k)∈N (otherwise identical nodes will appear several times in the graph).

The graph is built fromn₀(t₀)ton_T(t_f)using the cor- responding edge sets. Algorithm 2 details the nodes building and the Figure 6 presents an example of graph building.

Algorithm 2:NODES Data:S,n₀(t₀),n_T(t_f) 1 L⁼ⁿ0(t₀),N=/0; 2 whileLis not emptydo 3 n_i(t_k) =L.pop out();

4 N.add(n_i(t_k));

5 ift_k<t_f then

6 forall[u]∈U(n_i(t_k))do 7 ni(t_k+1) ={si⊆S(t_k+1)|

ϕ(t_k,t_k+1;n_i(t_k),[u])∩s_i6=/0};

8 L^.add(ni(t_k+1));

9 N.add(n_T(t_f));

Result:N.

The computed graph corresponds to all the possible trajectories of the system that may lead to the target att_f from an initial state[x₀] att₀. A priori it en- closes the optimal trajectory. This graph has a partic- ularity: as the nodes of the graph are cell sets, it can be associated a reachability information for each node n_i(t_k)∈N

Figure 6: Example of graph building. Starting from a node n0(t0) ={si⊆S(t0)|si∩[x0]6= /0}, two nodes are com- puted:n1(t1) ={si⊆S(t1)|si∩[x1]6=/0}andn2(t1) ={si⊆ S(t1)|si∩[x2]6= /0}, with [x1] =ϕ(t0,t1;n0(t0),[u1]) and [x2] =ϕ(t0,t1;n0(t0),[u2]). The same principle is repeated for the other nodes. It can be noticed that

--- the top figure corresponds to the superimposition of the three slicesS(t₀),S(t₁), andS(t₂),

- some cells can belong to different nodes, as the dark grey cell is attached to the noden₂₂(t₂)andn₂₁(t₂).

- a noden_i(t_k)isreachableif all the cellss_i∈n_i(t_k) arereachable,

- a noden_i(t_k)isindeterminateif at least one cell s_i∈n_i(t_k)is notreachable.

This can be extended to the paths of the graph - a path isreachableif all the nodes of the path are

reachable,

- a path isindeterminateif at least one node of the path isindeterminate. It can be noticed that an indeterminatepath may correspond to a trajectory that does not exist considering the system. They have to be considered carefully.

5.3 Exploitation of the graph

Using this graph, it is possible, with a shortest path algorithm, to compute two informations:

- an enclosure of the optimal control vector to reach the targetTfrom an initial state[x₀]∈C_t0,tf, - an evaluation of the cost of this control vector.

The chosen shortest path algorithm is a general- ization of the Dijkstra algorithm (Dijkstra, 1971).

The classical Dijkstra algorithm is presented Algo- rithm 3. The input of this algorithm is a graphG,

composed by a set of nodes N linked to each oth- ers with weighted edges. J(n_i) corresponds to the weight of the node n_i and J(n_i,n_j) corresponds to the weight of the edge linking the nodesn_i andn_j (in our case it corresponds to the cost of the control vector from n_i to n_j). The Dijkstra algorithm can easily be extended for edges with interval weights as the min() function can be extended to intervals (min([x₁,x₁],[x₂,x₂]) = [min(x₁,x₂),min(x₁,x₂)] e.g.

min([3,9],[5,7]) = [3,7]). Note that with interval weights it may not be possible to choose between two paths, in this case, both paths are solutions.

Algorithm 3:Dijkstra Algorithm Data:G

1 initialize the nodes asunmarked;

2 ∀n_i∈N,J(n_i) = +∞; 3 L(n₀) =0;

4 whileit exists an unmarked nodedo

5 n_L= the unmarked node with the lowestJ;

6 noten_Lasmarked;

7 forall unmarked nodes nUlinked to nLdo 8 J(n_U) =min(J(n_U),J(n_L) +J(n_L,n_U));

Result: weighted node setN.

As some paths may not correspond to possible trajec- tories of the studied system (indeterminatepaths), a reachablesub-graph has to be considered. Be a graph G, it is possible to build a sub-graphG_rdefined by all thereachablepaths ofG. In this caseGcorresponds to all the trajectories that may lead the system to the target andG_rcorresponds to all the guaranteed trajec- tories that lead the system the target from the initial state.

Processing an interval Dijkstra algorithm overG_rit is possible to find a setP^∗_r of shortest guaranteed paths for this sub-graph.

P^∗_r ={shortest pathsP∈G_r} (24) As mentioned before, when it is not possible to deter- minate if a path is shorter than an other one the two paths have to be kept as solution. For example if a pathP₁has a costJ(P₁) = [20,25]and a pathP₂has a costJ(P₂) = [22,30], it is not possible to determinate which path is shorter (J(P₁)6≮J(P₂)since 25>22, and J(P₂)6≮J(P₁)since 22>20). In this case the two pathsP₁andP₂have to be kept as they may both be the shortest path.

The paths so obtained are guaranteed to exist, but may not be the shortest paths considering the graphG. The idea is to considerP^∗_ras an upper bound of the shortest path ofG.

KnowingP^∗_r and processing an other interval Dijkstra algorithm overGit is possible to finally find the path setP^∗that encloses the shortest path ofG:

P^∗={pathsP∈Gthat may be better thanP^∗_r} ∪P^∗_r (25) Note that a path may be better than an other, if it is not possible to determinate which path is shorter.

Figure 7: A graph example. Each node corresponds to a set of cells. The edges have interval weights corresponding to the costsJ([u])of the control vectors[u]∈U. Note that n₀,n₁,n₂,n₁₂,n₂₁,n₁₂₁,n₂₁₁ arereachable nodes whereas n₁₁,n₁₁₁,n₂₁₂areindeterminatenodes.

Example of the Figure 7

Considering the graphGof the Figure 7, the following reachablesub-graph can be computed:

- N_r = {n₀,n₁,n₂,n₁₂,n₂₁,n₁₂₁,n₂₁₁,n_T} corre- sponding to thereachablenodes and edges, withN_rthe node set ofG_r.

For the reader information here are the computation of all the path costs:

- J(P(n₀,n₁,n₁₁,n₁₁₁,n_T)) = [6,10]

- J(P(n₀,n₁,n₁₂,n₁₂₁,n_T)) = [11,15]

- J(P(n₀,n₂,n₂₁,n₂₁₁,n_T)) = [5,9]

- J(P(n₀,n₂,n₂₁,n₂₁₂,n_T)) = [14,18]

Using the interval Dijkstra algorithm it is possi- ble to evaluate the optimal paths of the graph G_r, P^∗_r = {P(n₀,n₂,n₂₁,n₂₁₁,n_f)}. By process- ing an other interval Dijkstra algorithm over G and keeping only the paths that may be better that P^∗_r we obtain P^∗={P(n₀,n₁,n₁₁,n₁₁₁,n_T)} ∪ P^∗_r. It can be concluded that the optimal path of the system for this example is included in P^∗={P(n₀,n₁,n₁₁,n₁₁₁,n_f),P(n₀,n₂,n₂₁,n₂₁₁,n_T)} and has a costJ(P^∗) = [5,9].

6 CONCLUSION

In this paper, we have introduced two interval-based algorithms. The first one allows to compute the sub- setC_t0,tf of initial states from which there exists at least one trajectory of the system reaching the target Tin finite timet_f from a timet₀. The result of this work is an outer and inner characterisation ofC_t0,tf. Then adapting this work we have defined an other al- gorithm to deal with optimal control characterization.

This second algorithm computes a graph correspond- ing to all the possible trajectories that my lead the sys- tem from an initial state to the target. Using a gen- eralized Dijkstra algorithm, it is possible to use this graph in order to enclose the optimal control vector and evaluate is cost.

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