![](bg2.png)
is used to compute the approximations, such that
C
−
t
0
,t
f
⊆ C
t
0
,t
f
⊆ C
+
t
0
,t
f
, by using guaranteed numeri-
cal integration (VNODE-LP
1
). Note that an obvious
approximations would be C
−
t
0
,t
f
=
/
0 and C
+
t
0
,t
f
= K.
The proposed method aims at computing a better en-
closure of C
t
0
,t
f
.
In a second part, we got interested to the optimal
control problem. Given a cost function J and an ini-
tial state x
0
∈ C
t
0
,t
f
, we propose a numerical method
to evaluate an enclosure of the discrete optimal con-
trol u(t) ∈ U such that ϕ(t
0
,t
f
;x
0
,u(t)) ∈ T and u(t)
continuous over [t
k
,t
k+1
].
The paper is organised as follow. First some in-
terval analysis tools are presented in Section 2 as they
are used to compute the inner and outer approxima-
tions. Section 3 presents the proposed algorithm to
compute C
t
0
,t
f
and is followed by experimental re-
sults in Section 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 denoted IR. An in-
terval is usually denoted using brackets. An element
of an interval [x] is denoted by x. An interval vector
(box) [x] of R
n
is a Cartesian product of n intervals. If
[x] = [x
1
,x
1
] ×···×[x
n
,x
n
] is a box, then its width is
w([x]) = w([x
1
]) ×···×w([x
n
]), (7)
where w([x
i
]) = x
i
−x
i
. The set of all boxes of R
n
is
denoted by IR
n
.
The Bisect() function divides an interval [x] into two
intervals [x
1
] and [x
2
] such as [x
1
] ∪[x
2
] = [x], [x
1
] ∩
[x
2
] =
/
0 and w([x
1
]) = w([x
2
]).
The main concept of interval analysis is the extension
of real functions to intervals, which is defined as fol-
lows. Let f : R
n
→ R
m
be a continuous real function,
and [f] : IR
n
→ IR
m
be an inclusion function. Then
[f] is an inclusion function of f if and only if for every
[x] ∈ IR
n
,{f(x)|x ∈ [x]} ⊆ [f]([x]).
Hence, an interval inclusion allows computing en-
closures of the image of boxes by real functions. It
1
A C++ package for computing bounds on solutions in Initial
Value Problems for Ordinary Differential Equations, by N. Nedi-
alkov.
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 −,×,÷,x
n
,
√
x,exp,··· This process gives rise
to the so-called interval 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.
Interval arithmetic can be used to compute guaranteed
integration. In the later, the Nedialkov method is used
to compute:
- [x]
∗
such that [x]
∗
⊃ ϕ(t
k
,t
k+1
;[x],[u
k
]),
- K
∗
such that K
∗
⊃ φ([t
k
,t
k+1
];[x],[u
k
]).
Note that the Nedialkov method is one chosen solu-
tion over several methods, one could chose a different
approach.
Given a bounded set E of complex shape, one usu-
ally defines an axis-aligned box or paving, i.e. an
union of non-overlapping boxes, E
+
which contains
the set E : this is known as the outer approximation of
it. Likewise, one also defines an inner approximation
E
−
which is contained in the set E. Hence we have
the following property
E
−
⊆ E ⊆E
+
(8)
3 CHARACTERIZATION OF C
T
0
,T
F
This section presents an algorithm able to provide an
inner and an outer approximation of C
t
0
,t
f
assuming
that the input u(t
k
) is continuous over [t
k
,t
k+1
], and
boundedso it is possible to determinatea box[u
k
] such
that u(t) ∈ [u
k
] over [t
k
,t
k+1
]. That is, the obtained
results will be dependant of the time’s step δ
t
.
For each time t
k
the algorithm computes a gridding of
K (a slice), noted S(t
k
). The resolution of the gridding
is δ
K
= (δ
x
1
,··· ,δ
x
i
,··· ,δ
x
n
) where δ
x
i
corresponds
to the resolution of the i
th
dimension of K (Figure 1).
A cell s
i
of S(t
k
) can be
- unreachable if no state x in this cell allows the
system to reach the target at time t
f
, for all possi-
ble input vectors. The set of all the unreachable
cells of S(t
k
) is noted S
u
(t
k
)
S
u
(t
k
) = { s
i
∈S(t
k
)|∀u(t) ∈ U,
φ([t
k
,t
f
];s
i
,u(t)) ∩T =
/
0}
(9)
- reachable if for all the states x of this cell it exists
an input vector that allows the system to reach the
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