While I am familiar with Descartes and the oft-quoted idea, I think therefore I am, I am not familiar with any philosophy that actively sought to rationalize that anybody else exists.
Are there philosophers who analyzed this question? Or is it taken for granted that it is (im)possible for anybody to validate their own existence to somebody else?
Philosophy's name for the idea you are describing is solipsism. The Wikipedia article defines solipsism as "the epistemological or ontological position that knowledge of anything outside one's own specific mind is unjustified".
The IEP article, Solipsism and the Problem of Other Minds, adequately summarizes the cases for and against solipsism. The article makes reference to relevant works by philosophers including Descartes, Locke, Wittgenstein, Ayer, Quine, Mill and others. The passage from the article which is most relevant to your specific question is entitled The Incoherence of Solipsism and has been copied below:
With the belief in the essential privacy of experience eliminated as false, the last presupposition underlying solipsism is removed and solipsism is shown as foundationless, in theory and in fact. One might even say, solipsism is necessarily foundationless, for to make an appeal to logical rules or empirical evidence the solipsist would implicitly have to affirm the very thing that he purportedly refuses to believe: the reality of intersubjectively valid criteria and a public, extra-mental world. There is a temptation to say that solipsism is a false philosophical theory, but this is not quite strong or accurate enough. As a theory, it is incoherent. What makes it incoherent, above all else, is that the solipsist requires a language (that is a sign-system) to think or to affirm his solipsistic thoughts at all. Given this, it is scarcely surprising that those philosophers who accept the Cartesian premises that make solipsism apparently plausible, if not inescapable, have also invariably assumed that language-usage is itself essentially private. The cluster of arguments – generally referred to as “the private language argument” – that we find in the Investigations against this assumption effectively administers the coup de grâce to both Cartesian dualism and solipsism. (I. § 202; 242-315). Language is an irreducibly public form of life that is encountered in specifically social contexts. Each natural language-system contains an indefinitely large number of “language-games,” governed by rules that, though conventional, are not arbitrary personal fiats. The meaning of a word is its (publicly accessible) use in a language. To question, argue, or doubt is to utilize language in a particular way. It is to play a particular kind of public language-game. The proposition “I am the only mind that exists” makes sense only to the extent that it is expressed in a public language, and the existence of such language itself implies the existence of a social context. Such a context exists for the hypothetical last survivor of a nuclear holocaust, but not for the solipsist. A non-linguistic solipsism is unthinkable and a thinkable solipsism is necessarily linguistic. Solipsism therefore presupposes the very thing that it seeks to deny. That solipsistic thoughts are thinkable in the first instance implies the existence of the public, shared, intersubjective world that they purport to call into question.
The analysis and arguments from IEP summarize the approach to the issue in analytic philosophy. [Continental philosophy], specifically phenomenology, addresses this skepticism by introducing the concept of intersubjectivity. Intersubjectivity is based on a phenomenlogical ontology, which "is the study of structures of consciousness as experienced from the first-person point of view". The phenomenological equivalent to "I think, therefore, I am" is "I apprehend myself as a subject of consciousness" (by contrast to an object). Intersubjectivity is basically the apprehension of an Other as a subject (like the self), and its existence is the refutation of solipsism. This refutation requires the acceptance of a phenomenological paradigm.
Existentialism (based on phenomenlogical ontology) has sometimes been accused of espousing solipsism. However, Sartre argues that existentialism and solipsism are necessarily incompatible. The most concise and easily digestible instance of this argument can be found in the text of his 1964 talk, Existentialism Is A Humanism.
In an effort to "to offer a defence of existentialism against several reproaches that have been laid against it", Sartre describes a common mistake about existentialism as:
Both from this side and from the other we are also reproached for leaving out of account the solidarity of mankind and considering man in isolation. And this, say the Communists, is because we base our doctrine upon pure subjectivity – upon the Cartesian “I think”: which is the moment in which solitary man attains to himself; a position from which it is impossible to regain solidarity with other men who exist outside of the self. The ego cannot reach them through the cogito.
There has been quite a bit of work done "that actively sought to rationalize that anybody else exists". However, there haven't been any notable proponents of solipsism, (most likely because such a view contradicts our common sense).
Besides the reference to Solipsism (in another answer here), you may want to search a bit on the "Problem of Other Minds", which is the name that this problem usually goes by in the literature. (cf. http://plato.stanford.edu/entries/other-minds/)
The short version is, there are a number of philosophers who have treated the problem of other minds from a variety of perspectives. I recall reading a nice book on the subject by Simon Glendinning, who compared the approaches of Derrida, Wittgenstein and MacDowell.
