Can I find the encryption key if I know the plain text and the encrypted text (DES and AES)? If I have the plain text and its output after encryption with a key K1, is it algorithmically feasible to find K1?

I am specifically interested in the cases of DES and AES encryption algorithms.
The most cost-effective way to do this is to try each key and see if it works.  The expected number of trials is $2^{55}$ with DES, $2^{127}$ with AES-128, and $2^{255}$ with AES-256.

You can speed the search up by making many trials in parallel, e.g. by using a million DES circuits making one trial every microsecond, which will give an answer in a few hours, at most under a day—a task that has even been commercialized (note: I do not endorse this commercial service; I am only mentioning it for practical scale comparison).  Note that parallelizing the search doesn't change the overall cost: powering a single circuit for an hour will cost approximately the same energy, and same number of rubles, as powering sixty circuits in parallel for a minute.

If you have many keys $k_i$ which you want to recover any one of given their respective ciphertexts for a common plaintext, then you can do better with a parallel machine that saves repeated effort using Oechslin's rainbow tables.  Say you have $n$ target keys.  A machine parallelized $n^2$ ways with each parallel circuit not much larger than an AES-128 circuit, with a small constant amount of memory for each circuit, can give you your first answer with probability $p$ in about $2^{128}p/n^3$ time, at the same cost as about $2^{128}p/n$ AES-128 evaluations.

For AES-128, these numbers are large, but not unimaginably so.  The Bitcoin network today performs about $30\times10^{18}$ SHA-256 computations every second, about $2^{64}$ per second or $2^{90}$ per year.  If we had trillion ($10^{12}$) AES-128 circuits in parallel attacking any one of a million ($10^6$) keys, we would break at least one of them with high probability using a couple hundred thousand times the annual cost of the Bitcoin network (which has been increasing exponentially since about 2015), assuming the cost of one AES-128 evaluation is approximately the same as that of SHA-256.

Can you break AES-128, or any generic 128-bit function, using this method?  No.  Can humanity?  Maybe.  But definitely not AES-256 unless there's a cryptanalytic breakthrough.  (Quantum computers don't make much of a dent in AES-256 or even AES-192 either.)