239
of the efforts made thus far, the evidence seems to indicate that the aim is an attainable
one.
The main goal of this section is to carry out a theoretical study on the remainig part
of the von Neumann programme: the control of the climate. Our modest goal is to study
such a general philosophy by considering the simple climate models introduced by M.l.
Budyko and W. D. Sellers.
Continuing our previous research (see Diaz [1994]) in which it was shown how the
obstruction phenomenon leads to the general uncontrollability of the Sellers model, we
show here that a chance still remains: the restricted (approximate) controllability. We
will show that a very large class of desired climate states are attainable (in a weak sense)
by introducing suitable spatially localized controls on the climate system.
Our main goal is to study if possible antropogenerated actions on the climate system
allows to carry the average temperature from a given distribution y(O, x) to a desired
distribution Yd( x) after a given period of time T. Such type of questions was already
considered by J. Fourier [1824J and some of the most relevant applied mathematicians
of this century (J. von Neumann [1955J and J.L. Lions [1990J [1992J among them). The
connection between this question and the study of the irreversibility of the antropogenetic
changes already introduced in the atmosphere since the beginings of the Industrial Society
is obvious. It is also clear that many of the actual world decissions on greenhouse gases
emmision norms follow also this philosofy.
A mathematical statement of the question under consideration can be the following:
given w an open submanifold of M, T > 0, an initial distribution of temperatures Uo :
M --+ JR and a desired temperature Yd : M --+ JR, we want to find a control v : (0, T) xw --+
JR such that y(T : v) = Yd where y(. : v) denotes the solution of problem (P) replacing
f(t,x) by f(t,x) + v(t,x)Xw with Xw the characteristic function of w. When the answer
is positive we say that (P) is controllable. Nevertheless, the parabolic character of the
equation of (P) implies some regularizing effects making impossible our goal except for
a very limited class of desired states Yd. A relaxed statement comes in a natural way:
the approximate controllability. Given E > ° we seek now a control v. (defined again on
(0, T) x w) such that d(y(T, v.), Yd) ::; E. In the above expression d(·,·) represents the
distance measured in some space of functions defined on M (usually £2(M), or, more
generally, LP(M) with 1::; p::; (0).
The nature of our spatial domain M leads to some additional (and technical) difficulties in our study. A simpler formulation which still gives a representative idea of the
treatement in more complex situations corresponds to the case in which we replace M
by an open regular bounded set n of JR2 (here JR2 can be also substitued by JRN with
N E IN). As boundary condition on (0, T) x an we can chose the one of Neumann type
since it leads to a set of test functions for the weak formulation very similar to the one
corresponding to the case of a Riemannian manifold without boundary. Another unrele-
of the efforts made thus far, the evidence seems to indicate that the aim is an attainable
one.
The main goal of this section is to carry out a theoretical study on the remainig part
of the von Neumann programme: the control of the climate. Our modest goal is to study
such a general philosophy by considering the simple climate models introduced by M.l.
Budyko and W. D. Sellers.
Continuing our previous research (see Diaz [1994]) in which it was shown how the
obstruction phenomenon leads to the general uncontrollability of the Sellers model, we
show here that a chance still remains: the restricted (approximate) controllability. We
will show that a very large class of desired climate states are attainable (in a weak sense)
by introducing suitable spatially localized controls on the climate system.
Our main goal is to study if possible antropogenerated actions on the climate system
allows to carry the average temperature from a given distribution y(O, x) to a desired
distribution Yd( x) after a given period of time T. Such type of questions was already
considered by J. Fourier [1824J and some of the most relevant applied mathematicians
of this century (J. von Neumann [1955J and J.L. Lions [1990J [1992J among them). The
connection between this question and the study of the irreversibility of the antropogenetic
changes already introduced in the atmosphere since the beginings of the Industrial Society
is obvious. It is also clear that many of the actual world decissions on greenhouse gases
emmision norms follow also this philosofy.
A mathematical statement of the question under consideration can be the following:
given w an open submanifold of M, T > 0, an initial distribution of temperatures Uo :
M --+ JR and a desired temperature Yd : M --+ JR, we want to find a control v : (0, T) xw --+
JR such that y(T : v) = Yd where y(. : v) denotes the solution of problem (P) replacing
f(t,x) by f(t,x) + v(t,x)Xw with Xw the characteristic function of w. When the answer
is positive we say that (P) is controllable. Nevertheless, the parabolic character of the
equation of (P) implies some regularizing effects making impossible our goal except for
a very limited class of desired states Yd. A relaxed statement comes in a natural way:
the approximate controllability. Given E > ° we seek now a control v. (defined again on
(0, T) x w) such that d(y(T, v.), Yd) ::; E. In the above expression d(·,·) represents the
distance measured in some space of functions defined on M (usually £2(M), or, more
generally, LP(M) with 1::; p::; (0).
The nature of our spatial domain M leads to some additional (and technical) difficulties in our study. A simpler formulation which still gives a representative idea of the
treatement in more complex situations corresponds to the case in which we replace M
by an open regular bounded set n of JR2 (here JR2 can be also substitued by JRN with
N E IN). As boundary condition on (0, T) x an we can chose the one of Neumann type
since it leads to a set of test functions for the weak formulation very similar to the one
corresponding to the case of a Riemannian manifold without boundary. Another unrele-